Cool with statistical independence

# Cool with statistical independence
## Basic bivariate statistical tests (and false positives)
### Gina Reynolds
### September/October, 2019

---

---
# Statistical independence

> In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

> Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

---
## "A statistically significant relationship"

- When you have *evidence* of statistical *dependence*.  The probability of observing of observing such a strong relationship --- or stronger --- under *statistical independence* (often the null) is quite small.

https://www.youtube.com/watch?v=gSyGVDMcg-U

---
## Hypotheses

Hypothesis testing:

- A null hypothsis: Ho (Often statistical *independence*)
- Alternate hypothesis: Ha (Statistical *dependence*)

---
## Rejecting the null hypothesis

- Comparison will be made in relation to what we would expect to observe *under the null* (no relationship).  Such a strong relationship is unlikely to be observed by chance in a sample, if there is *not* dependence.

---

- If data *deviates* from expecations about null

- we reject the null (there is no relationship - statistical independence)
   - and accept the alternative (there is a relationship - statistical dependence)

---
## P-Value

The probability that a relationship as strong or stronger would have been observed under the null.

http://fivethirtyeight.com/features/not-even-scientists-can-easily-explain-p-values/ - but they all can...

---
## Fisher's exact test

```r
number_correct = rep(NA, 1000000)
for (i in 1:1000000){
  guess <- sample(c(rep("tea first", 4), rep("milk first", 4)))
  truth <- sample(c(rep("tea first", 4), rep("milk first", 4)))
  number_correct[i] = sum(guess == truth)
}

table(number_correct)

table(number_correct)/10000
```

---

# Two statistically *independent* continuous variables

---

```r
*library(tidyverse)
*set.seed(98735)
*tibble(tail_length = rnorm(n = 50),
*      brain_weight = rnorm(n = 50)) ->
*my_data; my_data
```
]
 
.panel2-continuous0-user[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -1.64       -0.530 
 2      -0.107      -0.744 
 3       0.185       0.476 
 4       1.36        1.32  
 5       0.210      -0.0852
 6      -0.658      -1.36  
 7       1.53       -1.18  
 8       1.40       -0.867 
 9       1.54        0.115 
10       0.131       0.960 
# … with 40 more rows
```
]

---
count: false

```r
library(tidyverse)
set.seed(98735)
tibble(tail_length = rnorm(n = 50),
       brain_weight = rnorm(n = 50)) ->
my_data; my_data

*ggplot(data = my_data) +
* aes(x = tail_length) +
* aes(y = brain_weight) +
* geom_point(color = "steelblue",
*            alpha = .8)
```
]
 
.panel2-continuous0-user[

![](statistical_independence_files/figure-html/continuous0_user_02_output-1.png)
]

---
count: false

```r
library(tidyverse)
set.seed(98735)
tibble(tail_length = rnorm(n = 50),
       brain_weight = rnorm(n = 50)) ->
my_data; my_data

ggplot(data = my_data) +
  aes(x = tail_length) +
  aes(y = brain_weight) +
  geom_point(color = "steelblue",
             alpha = .8)
*cor.test(x = my_data$tail_length,
*             y = my_data$brain_weight)
```
]
 
.panel2-continuous0-user[

![](statistical_independence_files/figure-html/continuous0_user_03_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.14569, df = 48, p-value = 0.8848
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2976302  0.2588382
sample estimates:
        cor 
-0.02102428 
```
]

---

```r
library(tidyverse)
tibble(tail_length = rnorm(n = 50),
       brain_weight = rnorm(n = 50)) ->
my_data; my_data
#
#
#
#
#
#
#
#
#
#
#
#
#
ggplot(data = my_data) +
  aes(x = tail_length) +
  aes(y = brain_weight) +
  geom_point(color = "steelblue",
             alpha = .8) +
  theme(panel.background =
          element_rect(color =
                         ifelse(cor.test(my_data$tail_length, my_data$brain_weight)[[3]]<.05, "red", "black"),
                       size =
                         ifelse(cor.test(my_data$tail_length, my_data$brain_weight)[[3]]<.05, 8, 3)))

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1       1.14        0.271 
 2       0.252       0.359 
 3       1.27        0.704 
 4      -0.332      -3.60  
 5      -0.555       0.0577
 6       0.632      -0.812 
 7       0.202      -0.206 
 8      -0.819      -0.866 
 9       1.08        0.319 
10       0.403      -1.62  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_01_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 0.15018, df = 48, p-value = 0.8812
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2582334  0.2982208
sample estimates:
       cor 
0.02167209 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      0.555        0.689 
 2      1.37         0.408 
 3     -1.20         0.0598
 4     -0.824        0.0650
 5      0.398        0.109 
 6     -0.713       -0.918 
 7      0.267        0.293 
 8      0.0750       0.308 
 9      0.231       -0.183 
10     -0.837        0.868 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_02_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.53715, df = 48, p-value = 0.5936
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3481556  0.2054699
sample estimates:
        cor 
-0.07729874 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     -1.93          0.591
 2      1.37          1.60 
 3     -0.728         0.625
 4     -0.0645       -0.812
 5      0.297        -0.337
 6     -0.912        -1.63 
 7     -1.13         -1.09 
 8     -2.04         -0.756
 9     -1.32          1.28 
10      0.409         2.17 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_03_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 1.073, df = 48, p-value = 0.2886
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.1308697  0.4137734
sample estimates:
      cor 
0.1530533 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -0.231      0.291  
 2      -0.575      0.113  
 3       0.251     -1.28   
 4      -1.38      -3.90   
 5       0.129      0.405  
 6       0.677     -1.29   
 7      -0.532     -1.13   
 8      -0.699     -0.00250
 9      -1.51      -0.845  
10       0.992     -0.560  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_04_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 2.5832, df = 48, p-value = 0.01289
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.07865544 0.57207274
sample estimates:
      cor 
0.3493545 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     -1.25          0.429
 2      0.754         0.143
 3     -0.0323       -0.173
 4      0.353        -0.931
 5      0.872         0.226
 6     -0.963         0.293
 7      0.178         0.890
 8      0.694        -2.32 
 9     -0.717         0.470
10      0.866        -1.29 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_05_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 0.3823, df = 48, p-value = 0.7039
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2267286  0.3284075
sample estimates:
       cor 
0.05509622 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -0.390       0.383 
 2      -0.577       2.63  
 3       0.639       0.0420
 4      -0.921       0.0238
 5      -0.195       1.01  
 6       0.934       0.209 
 7      -1.55        0.261 
 8      -1.05        1.04  
 9       1.32        0.478 
10      -1.74       -1.21  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_06_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.83665, df = 48, p-value = 0.4069
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3853765  0.1639290
sample estimates:
       cor 
-0.1198892 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -1.29        1.83  
 2      -0.538      -0.658 
 3      -0.439       0.410 
 4       0.884       0.383 
 5      -0.826       0.593 
 6      -0.645      -0.643 
 7      -0.277       0.526 
 8       1.33       -0.0301
 9      -1.26        2.05  
10      -0.329      -1.18  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_07_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.43527, df = 48, p-value = 0.6653
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3351996  0.2194761
sample estimates:
        cor 
-0.06270212 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     1.73          -0.403
 2     0.769          0.460
 3     0.625         -1.42 
 4     0.251          0.940
 5    -0.139         -0.347
 6     1.85          -1.04 
 7    -0.538          0.448
 8     1.14           0.727
 9    -0.00658       -2.39 
10     0.226          0.783
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_08_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.95076, df = 48, p-value = 0.3465
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3991969  0.1479922
sample estimates:
      cor 
-0.135956 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     -1.36         0.237 
 2     -0.674       -1.29  
 3      0.820        0.439 
 4      0.126       -1.36  
 5      0.531       -0.706 
 6     -0.255       -0.168 
 7      0.620       -0.152 
 8      0.0415      -0.0826
 9      0.977       -1.27  
10      1.88         1.38  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_09_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.038026, df = 48, p-value = 0.9698
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2834033  0.2732767
sample estimates:
         cor 
-0.005488532 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     -0.785        0.111 
 2      0.674        1.56  
 3      0.0216       0.976 
 4      0.223        1.04  
 5     -0.575       -0.0359
 6      0.796       -1.27  
 7     -2.24        -0.921 
 8      0.817        1.73  
 9     -2.11         2.84  
10      0.453       -1.24  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_10_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 0.53597, df = 48, p-value = 0.5945
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2056319  0.3480069
sample estimates:
       cor 
0.07713057 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      0.631      -0.544  
 2      0.236      -1.47   
 3     -0.583       1.90   
 4     -0.148      -0.00957
 5     -0.616       1.11   
 6      1.08       -1.39   
 7      0.325       0.557  
 8     -1.24        0.323  
 9     -0.0400     -0.637  
10     -0.726       1.64   
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_11_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -1.9265, df = 48, p-value = 0.05998
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.50834031  0.01129057
sample estimates:
       cor 
-0.2678991 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -0.470      -0.470 
 2      -1.46       -0.600 
 3      -0.134      -0.0555
 4      -1.39        0.183 
 5       0.874      -1.34  
 6      -1.12       -1.46  
 7      -0.208       1.53  
 8      -1.01       -0.175 
 9      -0.209       0.402 
10      -0.222      -0.242 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_12_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 1.3943, df = 48, p-value = 0.1697
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.08576768  0.45087766
sample estimates:
    cor 
0.19729 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     -1.55        1.33   
 2      0.320       0.882  
 3     -0.0949      0.483  
 4      0.756      -1.69   
 5      0.804      -0.390  
 6     -0.251       1.00   
 7     -1.86        0.358  
 8      1.06       -0.00587
 9      0.912      -0.265  
10     -2.25       -0.117  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_13_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 0.52319, df = 48, p-value = 0.6032
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2073927  0.3463893
sample estimates:
       cor 
0.07530195 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -0.491       0.554 
 2      -0.351      -0.724 
 3       0.995      -0.559 
 4      -1.80        0.949 
 5      -0.798       0.337 
 6       0.582      -0.0307
 7      -0.415      -0.574 
 8      -1.07        1.24  
 9       1.13       -0.163 
10      -0.788       0.201 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_14_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 1.5728, df = 48, p-value = 0.1223
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.06071377  0.47071754
sample estimates:
      cor 
0.2213751 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -0.745       0.0542
 2      -0.625      -0.667 
 3       0.343      -0.405 
 4      -0.672       1.08  
 5       0.631      -0.909 
 6      -0.312      -0.718 
 7       0.796      -0.437 
 8       0.712       0.141 
 9      -1.28        0.238 
10       0.181       0.0345
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_15_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.43976, df = 48, p-value = 0.6621
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3357731  0.2188609
sample estimates:
        cor 
-0.06334581 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -1.24         0.123
 2      -1.61         0.382
 3       0.511        0.507
 4      -0.593        0.637
 5      -1.15        -0.474
 6       0.228        1.22 
 7      -0.232       -0.509
 8      -0.102       -0.244
 9       0.367       -1.80 
10      -0.300        0.101
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_16_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = 0.25132, df = 48, p-value = 0.8026
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2445639  0.3114566
sample estimates:
       cor 
0.03625162 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1       1.33         1.17 
 2       1.09         0.724
 3      -0.651       -1.40 
 4      -1.05         1.61 
 5      -0.334       -1.50 
 6      -0.216       -2.62 
 7       0.839       -0.204
 8      -1.34         2.19 
 9      -0.578        0.490
10      -1.42        -0.466
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_17_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.22808, df = 48, p-value = 0.8206
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3084259  0.2477135
sample estimates:
        cor 
-0.03290287 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     -2.18          0.751
 2     -0.254         0.185
 3     -0.484         0.471
 4      0.742        -0.597
 5      1.45          0.672
 6      1.56          1.18 
 7      1.34          2.17 
 8      2.67         -0.410
 9      0.877         0.266
10      0.0153        1.09 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_18_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.50468, df = 48, p-value = 0.6161
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3440425  0.2099411
sample estimates:
        cor 
-0.07265216 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1      -0.145       -0.873
 2       2.29         0.622
 3      -1.47        -0.162
 4      -0.285       -1.73 
 5      -0.598       -0.736
 6       0.782        0.662
 7       0.102       -0.312
 8       0.970       -1.42 
 9       0.422        0.370
10      -1.24         2.63 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_19_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -0.78761, df = 48, p-value = 0.4348
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.3793744  0.1707621
sample estimates:
       cor 
-0.1129545 
```
]

---
count: false

cor.test(x = my_data$tail_length,
              y = my_data$brain_weight)
```
]
 
.panel2-continuous-20[

```
# A tibble: 50 x 2
   tail_length brain_weight
         <dbl>        <dbl>
 1     -0.0717      -0.736 
 2      0.724        0.141 
 3     -0.562       -1.57  
 4     -0.189        0.808 
 5      0.957       -0.553 
 6     -0.489       -0.244 
 7      1.09         0.141 
 8      0.437       -0.872 
 9     -0.0162       0.0422
10     -2.04        -0.420 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/continuous_20_20_output-1.png)

```

Pearson's product-moment correlation

data:  my_data$tail_length and my_data$brain_weight
t = -1.4716, df = 48, p-value = 0.1477
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.45954213  0.07490899
sample estimates:
       cor 
-0.2077709 
```
]

---

```r
tibble(sex = sample(x = c("male","female"),
                  size = 50,
                  replace = T)) %>%
  mutate(height = rnorm(n = 50, sd = .5) + 5.5) ->
height_data; height_data
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
# visualization
ggplot(height_data) +
  aes(x = sex) +
  aes(y = height) +
  aes(group = sex) +
  geom_boxplot() +
  geom_jitter(height = 0,
              width = .02) +
  stat_summary(fun.y = mean,
               geom = "point",
               col = "goldenrod3",
               size = 5)

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   6.05
 2 female   5.76
 3 female   5.18
 4 female   4.40
 5 female   4.66
 6 male     6.07
 7 female   6.66
 8 female   5.78
 9 female   6.39
10 male     5.47
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_01_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = 0.17273, df = 46.923, p-value = 0.8636
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.277471  0.329592
sample estimates:
mean in group female   mean in group male 
            5.569354             5.543294 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   4.80
 2 male     5.75
 3 male     5.35
 4 female   5.43
 5 male     5.07
 6 male     5.88
 7 male     5.00
 8 female   4.07
 9 male     5.83
10 female   6.18
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_02_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.92503, df = 47.468, p-value = 0.3596
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.4019456  0.1486889
sample estimates:
mean in group female   mean in group male 
            5.414265             5.540893 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.00
 2 female   5.09
 3 female   5.05
 4 female   5.11
 5 male     4.95
 6 male     5.83
 7 female   5.28
 8 female   5.16
 9 male     5.42
10 male     5.52
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_03_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.69666, df = 46.762, p-value = 0.4895
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.3692719  0.1793219
sample estimates:
mean in group female   mean in group male 
            5.399995             5.494970 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   6.40
 2 male     5.95
 3 male     6.93
 4 female   5.58
 5 female   5.68
 6 female   5.46
 7 female   5.37
 8 male     5.43
 9 male     5.86
10 female   5.05
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_04_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -1.4178, df = 35.013, p-value = 0.1651
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.48788008  0.08663595
sample estimates:
mean in group female   mean in group male 
            5.428950             5.629572 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.04
 2 female   5.91
 3 male     5.05
 4 female   6.51
 5 male     5.42
 6 male     5.58
 7 female   5.67
 8 female   5.10
 9 male     5.68
10 female   5.52
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_05_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = 2.6639, df = 40.441, p-value = 0.01104
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.08008769 0.58298641
sample estimates:
mean in group female   mean in group male 
            5.671535             5.339998 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 male     5.84
 2 male     6.31
 3 female   5.65
 4 male     5.29
 5 male     5.80
 6 male     5.84
 7 female   5.72
 8 male     5.46
 9 female   5.50
10 male     5.72
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_06_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -1.0138, df = 47.933, p-value = 0.3158
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.4228784  0.1393819
sample estimates:
mean in group female   mean in group male 
            5.469518             5.611266 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 male     5.01
 2 female   5.68
 3 female   5.44
 4 female   5.10
 5 female   5.81
 6 male     5.19
 7 male     5.12
 8 male     6.21
 9 male     5.56
10 male     5.26
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_07_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -1.0251, df = 47.828, p-value = 0.3105
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.4501601  0.1461517
sample estimates:
mean in group female   mean in group male 
            5.567211             5.719215 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.66
 2 female   5.72
 3 female   5.63
 4 female   5.48
 5 male     5.47
 6 male     5.89
 7 male     6.00
 8 female   5.57
 9 male     5.41
10 male     4.77
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_08_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -1.58, df = 47.842, p-value = 0.1207
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.55526540  0.06662354
sample estimates:
mean in group female   mean in group male 
            5.347316             5.591637 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   4.23
 2 male     5.99
 3 male     5.43
 4 female   5.40
 5 male     6.16
 6 male     5.97
 7 female   5.78
 8 female   6.03
 9 female   5.52
10 female   5.15
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_09_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.80309, df = 41.183, p-value = 0.4265
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.4026129  0.1734895
sample estimates:
mean in group female   mean in group male 
            5.387418             5.501980 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   6.20
 2 male     5.04
 3 female   5.22
 4 female   5.69
 5 female   5.97
 6 female   5.64
 7 male     5.48
 8 female   5.64
 9 male     5.33
10 female   5.83
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_10_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.19366, df = 47.782, p-value = 0.8473
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.3081422  0.2540043
sample estimates:
mean in group female   mean in group male 
            5.500872             5.527941 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.02
 2 male     6.63
 3 male     5.35
 4 male     5.46
 5 male     5.53
 6 female   5.55
 7 male     6.33
 8 female   5.02
 9 male     5.49
10 male     6.09
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_11_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -2.0574, df = 47.703, p-value = 0.04513
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.569594465 -0.006505921
sample estimates:
mean in group female   mean in group male 
            5.491601             5.779651 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.47
 2 female   5.71
 3 male     6.34
 4 male     5.81
 5 male     4.99
 6 female   4.89
 7 female   6.02
 8 male     5.77
 9 male     5.47
10 male     5.42
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_12_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.025281, df = 48, p-value = 0.9799
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.2517547  0.2455024
sample estimates:
mean in group female   mean in group male 
            5.495759             5.498885 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.76
 2 female   4.89
 3 female   5.64
 4 male     5.47
 5 female   6.03
 6 male     6.47
 7 female   5.97
 8 female   5.50
 9 male     6.20
10 male     5.62
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_13_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.81926, df = 35.107, p-value = 0.4182
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.3814890  0.1620987
sample estimates:
mean in group female   mean in group male 
            5.550585             5.660281 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 male     5.74
 2 female   4.58
 3 male     5.47
 4 male     5.26
 5 male     5.01
 6 male     4.95
 7 female   5.86
 8 male     5.98
 9 female   5.51
10 male     6.12
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_14_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.39696, df = 47.977, p-value = 0.6932
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.3193852  0.2140676
sample estimates:
mean in group female   mean in group male 
            5.374303             5.426962 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 male     4.97
 2 male     5.02
 3 female   4.54
 4 female   5.16
 5 female   4.75
 6 female   5.71
 7 male     5.69
 8 male     6.76
 9 male     5.10
10 male     5.73
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_15_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -0.28826, df = 39.644, p-value = 0.7747
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.3477257  0.2609390
sample estimates:
mean in group female   mean in group male 
            5.460529             5.503923 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.11
 2 male     6.06
 3 female   5.21
 4 female   4.66
 5 male     5.61
 6 male     5.05
 7 male     5.36
 8 male     5.47
 9 male     5.43
10 male     5.42
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_16_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = 0.36447, df = 47.974, p-value = 0.7171
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.2070206  0.2986901
sample estimates:
mean in group female   mean in group male 
            5.447395             5.401561 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 male     6.16
 2 female   5.33
 3 male     6.05
 4 female   5.52
 5 female   5.36
 6 male     4.47
 7 male     5.40
 8 female   5.78
 9 female   5.38
10 male     4.96
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_17_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = 0.43687, df = 37.401, p-value = 0.6647
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.2420049  0.3751106
sample estimates:
mean in group female   mean in group male 
            5.467101             5.400549 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 male     5.36
 2 male     4.54
 3 male     5.56
 4 male     5.19
 5 female   5.13
 6 male     5.08
 7 male     6.45
 8 male     4.87
 9 male     4.93
10 female   5.59
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_18_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = 0.54448, df = 36.519, p-value = 0.5894
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.2020496  0.3504542
sample estimates:
mean in group female   mean in group male 
            5.442526             5.368323 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 male     5.44
 2 male     5.01
 3 female   5.36
 4 female   5.22
 5 female   5.10
 6 female   5.04
 7 female   6.61
 8 male     6.52
 9 female   5.36
10 female   5.35
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_19_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -1.5886, df = 43.93, p-value = 0.1193
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.51175262  0.06060527
sample estimates:
mean in group female   mean in group male 
            5.504991             5.730564 
```
]

---
count: false

# statistical test
t.test(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex)  # the discrete variable
```
]
 
.panel2-c_and_d2-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 female   5.18
 2 female   5.06
 3 male     5.97
 4 male     4.80
 5 female   5.91
 6 male     5.65
 7 male     4.61
 8 female   5.30
 9 female   5.17
10 female   5.36
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d2_20_20_output-1.png)

```

Welch Two Sample t-test

data:  height_data$height by height_data$sex
t = -1.6498, df = 45.988, p-value = 0.1058
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.45846363  0.04544978
sample estimates:
mean in group female   mean in group male 
            5.348123             5.554630 
```
]

---

# ANOVA (analysis of Variance)

---

```r
tibble(sex = sample(x = c("1st","2nd","3rd","Crew"),
                  size = 50,
                  replace = T)) %>%
  mutate(height = rnorm(n = 50)) ->
height_data; height_data
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
ggplot(height_data) +
  aes(x = sex) +
  aes(y = height) +
  aes(group = sex) +
  geom_boxplot() +
  geom_jitter(height = 0,
              width = .02) +
  stat_summary(fun.y = mean,
               geom = "point",
               col = "goldenrod3",
               size = 5)
#
#
#
#
#
#
#
#
#
#
#
#
# statistical test
TukeyHSD(aov(height_data$height ~ # the continuous variable by (~)  # the continuous variable by (~)
         height_data$sex))  # the discrete variable
```
]
 
.panel2-c_and_d3-20[

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 2nd    0.0646
 2 3rd   -0.280 
 3 3rd    1.01  
 4 2nd    1.21  
 5 3rd    0.358 
 6 1st    0.483 
 7 2nd    0.420 
 8 Crew   0.739 
 9 3rd   -1.01  
10 3rd    0.688 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_01_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
               diff        lwr       upr     p adj
2nd-1st  -0.1690278 -1.6428611 1.3048055 0.9899535
3rd-1st   0.1520384 -1.3724568 1.6765336 0.9933337
Crew-1st -0.2781325 -1.7785234 1.2222585 0.9600103
3rd-2nd   0.3210662 -0.6493905 1.2915229 0.8142269
Crew-2nd -0.1091047 -1.0412387 0.8230293 0.9893361
Crew-3rd -0.4301709 -1.4405049 0.5801631 0.6701215
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 Crew   0.184
 2 2nd   -0.549
 3 Crew   1.10 
 4 1st   -1.08 
 5 Crew  -0.236
 6 3rd    0.260
 7 2nd    0.336
 8 Crew   1.44 
 9 2nd    0.860
10 1st   -1.20 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_02_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
               diff        lwr       upr     p adj
2nd-1st   0.4340208 -0.6991114 1.5671529 0.7380964
3rd-1st   0.9033191 -0.1924056 1.9990438 0.1391734
Crew-1st  0.6144885 -0.4812362 1.7102132 0.4488395
3rd-2nd   0.4692983 -0.5718003 1.5103970 0.6290138
Crew-2nd  0.1804677 -0.8606310 1.2215664 0.9669081
Crew-3rd -0.2888306 -1.2890859 0.7114246 0.8676859
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 Crew  -1.79  
 2 Crew  -0.276 
 3 2nd    0.341 
 4 3rd    1.48  
 5 3rd   -0.942 
 6 3rd   -2.00  
 7 2nd   -0.0154
 8 2nd    0.243 
 9 Crew   0.127 
10 Crew   0.291 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_03_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff       lwr       upr     p adj
2nd-1st   0.02417633 -1.087864 1.1362170 0.9999291
3rd-1st  -0.21625886 -1.373707 0.9411898 0.9591147
Crew-1st -0.61673597 -1.834915 0.6014435 0.5368909
3rd-2nd  -0.24043519 -1.397884 0.9170134 0.9450743
Crew-2nd -0.64091230 -1.859092 0.5772672 0.5043507
Crew-3rd -0.40047711 -1.660245 0.8592904 0.8315205
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 3rd    1.00 
 2 3rd    0.506
 3 2nd    0.321
 4 3rd   -1.49 
 5 Crew   0.676
 6 Crew  -0.890
 7 3rd   -0.247
 8 2nd    0.278
 9 3rd    0.299
10 2nd    0.243
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_04_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr      upr     p adj
2nd-1st   0.31775146 -1.0543358 1.689839 0.9260481
3rd-1st   0.24293466 -0.9923450 1.478214 0.9528147
Crew-1st  0.51033616 -0.8617511 1.882423 0.7550174
3rd-2nd  -0.07481679 -1.2313865 1.081753 0.9981509
Crew-2nd  0.19258470 -1.1090916 1.494261 0.9789518
Crew-3rd  0.26740149 -0.8891682 1.423971 0.9263765
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 1st   -0.508 
 2 3rd   -0.794 
 3 2nd    0.469 
 4 2nd   -1.14  
 5 Crew   0.975 
 6 2nd    0.762 
 7 3rd   -1.69  
 8 Crew   0.0958
 9 Crew  -1.93  
10 3rd    0.190 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_05_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st   0.06671792 -1.0922126 1.2256484 0.9986935
3rd-1st  -0.08218738 -1.2549426 1.0905678 0.9976538
Crew-1st  0.46171866 -0.8162617 1.7396990 0.7709304
3rd-2nd  -0.14890529 -1.1485458 0.8507352 0.9785367
Crew-2nd  0.39500074 -0.7262308 1.5162323 0.7840762
Crew-3rd  0.54390603 -0.5916093 1.6794214 0.5821262
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 2nd   -0.569
 2 2nd    0.361
 3 1st   -0.292
 4 Crew  -0.483
 5 2nd   -0.160
 6 3rd    1.58 
 7 2nd    0.301
 8 3rd    0.224
 9 2nd   -0.427
10 2nd   -0.445
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_06_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
               diff        lwr       upr     p adj
2nd-1st  -0.4976309 -1.4783810 0.4831193 0.5350414
3rd-1st  -0.3340012 -1.3719295 0.7039272 0.8263871
Crew-1st -0.1880390 -1.2087153 0.8326373 0.9606999
3rd-2nd   0.1636297 -0.7352426 1.0625020 0.9620011
Crew-2nd  0.3095919 -0.5693030 1.1884868 0.7841348
Crew-3rd  0.1459622 -0.7963121 1.0882364 0.9759842
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 Crew  -0.0726
 2 3rd   -1.14  
 3 1st    0.540 
 4 3rd    0.0692
 5 1st   -0.481 
 6 3rd    1.32  
 7 2nd    0.153 
 8 3rd    0.197 
 9 2nd    0.120 
10 2nd   -0.0825
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_07_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st  -0.73135897 -1.5974410 0.1347231 0.1249395
3rd-1st  -0.28882958 -1.2317003 0.6540411 0.8463537
Crew-1st -0.03505996 -0.8660633 0.7959434 0.9994835
3rd-2nd   0.44252939 -0.5435484 1.4286072 0.6323153
Crew-2nd  0.69629901 -0.1834230 1.5760210 0.1652531
Crew-3rd  0.25376962 -0.7016454 1.2091846 0.8934204
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 Crew  -0.264
 2 2nd   -0.280
 3 1st   -0.191
 4 3rd   -0.154
 5 3rd   -0.902
 6 3rd   -1.61 
 7 3rd   -1.01 
 8 2nd    0.315
 9 3rd    0.762
10 1st    0.612
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_08_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st   0.16275305 -0.9710578 1.2965639 0.9807077
3rd-1st  -0.65568092 -1.7682972 0.4569353 0.4050145
Crew-1st -0.02476586 -1.2824543 1.2329226 0.9999473
3rd-2nd  -0.81843397 -1.9522448 0.3153769 0.2323237
Crew-2nd -0.18751891 -1.4639954 1.0889576 0.9793780
Crew-3rd  0.63091506 -0.6267734 1.8886035 0.5445379
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 Crew  -0.178 
 2 1st    0.332 
 3 3rd   -0.350 
 4 1st   -0.0232
 5 1st   -0.731 
 6 Crew  -1.50  
 7 2nd   -1.15  
 8 3rd    0.671 
 9 1st   -1.18  
10 3rd    0.281 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_09_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st   0.08690935 -0.9197195 1.0935382 0.9956444
3rd-1st   0.11918501 -0.8874438 1.1258139 0.9889722
Crew-1st -0.02914881 -1.1112423 1.0529447 0.9998652
3rd-2nd   0.03227565 -0.9928250 1.0573763 0.9997849
Crew-2nd -0.11605816 -1.2153561 0.9832398 0.9921183
Crew-3rd -0.14833382 -1.2476318 0.9509642 0.9838674
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 Crew  -0.709 
 2 3rd    1.27  
 3 Crew  -1.57  
 4 2nd   -0.763 
 5 1st   -0.0208
 6 1st   -0.784 
 7 3rd    0.644 
 8 3rd    1.23  
 9 2nd    0.224 
10 Crew   0.623 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_10_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff       lwr       upr     p adj
2nd-1st  -0.72308614 -1.792492 0.3463200 0.2853879
3rd-1st  -0.64836299 -1.780116 0.4833901 0.4300447
Crew-1st -0.88161058 -1.951017 0.1877955 0.1391813
3rd-2nd   0.07472315 -0.994683 1.1441293 0.9976743
Crew-2nd -0.15852444 -1.161716 0.8446674 0.9745747
Crew-3rd -0.23324759 -1.302654 0.8361585 0.9371878
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 Crew  -1.17 
 2 1st    2.08 
 3 3rd   -0.628
 4 1st   -0.872
 5 3rd    0.537
 6 1st   -2.35 
 7 3rd    1.02 
 8 2nd   -1.97 
 9 Crew   1.31 
10 1st    0.931
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_11_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st  -0.08727205 -1.1507634 0.9762194 0.9962523
3rd-1st  -0.16521339 -1.2287048 0.8982780 0.9757855
Crew-1st  0.17966865 -0.8179747 1.1773120 0.9631347
3rd-2nd  -0.07794134 -1.2034348 1.0475522 0.9977354
Crew-2nd  0.26694070 -0.7965507 1.3304321 0.9082050
Crew-3rd  0.34488204 -0.7186094 1.4083734 0.8230653
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 Crew   0.433 
 2 2nd   -0.0234
 3 2nd   -1.69  
 4 1st    0.0612
 5 1st    0.266 
 6 1st   -0.403 
 7 3rd   -0.762 
 8 Crew  -1.09  
 9 3rd   -0.0799
10 1st   -0.412 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_12_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st  -0.49469057 -1.2611645 0.2717834 0.3250737
3rd-1st  -0.03844713 -0.9414338 0.8645395 0.9994692
Crew-1st -0.15392662 -0.9355000 0.6276467 0.9526267
3rd-2nd   0.45624344 -0.4578916 1.3703785 0.5487530
Crew-2nd  0.34076396 -0.4536635 1.1351914 0.6649455
Crew-3rd -0.11547948 -1.0423115 0.8113525 0.9872030
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 2nd   -0.793 
 2 3rd    0.574 
 3 1st    0.479 
 4 3rd   -1.03  
 5 Crew   0.484 
 6 2nd    0.562 
 7 1st   -0.0160
 8 3rd    2.51  
 9 1st    0.0715
10 3rd    1.17  
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_13_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st   0.06107997 -1.0764645 1.1986244 0.9989386
3rd-1st  -0.06861325 -1.1088647 0.9716382 0.9980408
Crew-1st  0.49735172 -0.6027324 1.5974358 0.6268092
3rd-2nd  -0.12969321 -1.0666063 0.8072199 0.9826290
Crew-2nd  0.43627175 -0.5666580 1.4392015 0.6550294
Crew-3rd  0.56596497 -0.3250927 1.4570227 0.3389832
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 Crew   0.689
 2 Crew   1.05 
 3 2nd    0.769
 4 1st    0.302
 5 3rd    1.17 
 6 Crew   0.616
 7 3rd    0.435
 8 2nd   -0.143
 9 1st    1.10 
10 3rd    1.72 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_14_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
               diff       lwr       upr     p adj
2nd-1st  -0.2637570 -1.326638 0.7991240 0.9109553
3rd-1st   0.1261369 -1.010699 1.2629724 0.9908810
Crew-1st -0.4605313 -1.488838 0.5677751 0.6338422
3rd-2nd   0.3898939 -0.726890 1.5066778 0.7886407
Crew-2nd -0.1967743 -1.202868 0.8093198 0.9535344
Crew-3rd -0.5866682 -1.670598 0.4972619 0.4799025
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 2nd    1.03  
 2 1st   -0.306 
 3 3rd    0.939 
 4 3rd    1.04  
 5 2nd   -0.974 
 6 1st    0.0136
 7 Crew   0.117 
 8 1st    1.13  
 9 Crew  -0.444 
10 1st    0.0134
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_15_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
               diff        lwr       upr     p adj
2nd-1st   0.5442886 -0.4866955 1.5752728 0.5014093
3rd-1st   0.1250536 -0.7715174 1.0216246 0.9822438
Crew-1st  0.3688955 -0.8731682 1.6109593 0.8578628
3rd-2nd  -0.4192350 -1.3639132 0.5254432 0.6405627
Crew-2nd -0.1753931 -1.4526166 1.1018303 0.9830263
Crew-3rd  0.2438419 -0.9275722 1.4152560 0.9447585
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 2nd   -0.868 
 2 1st   -1.10  
 3 Crew  -0.887 
 4 3rd   -0.274 
 5 2nd   -2.02  
 6 1st   -1.20  
 7 3rd   -0.0952
 8 Crew   1.77  
 9 2nd    0.518 
10 2nd    0.373 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_16_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st  -0.71301214 -1.6919907 0.2659665 0.2253828
3rd-1st  -0.39014505 -1.3290118 0.5487217 0.6867008
Crew-1st -0.01296422 -0.9355957 0.9096673 0.9999808
3rd-2nd   0.32286709 -0.6379348 1.2836690 0.8070837
Crew-2nd  0.70004792 -0.2448957 1.6449915 0.2123895
Crew-3rd  0.37718083 -0.5261409 1.2805025 0.6834704
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 1st    1.24 
 2 3rd   -2.13 
 3 2nd    1.14 
 4 Crew  -0.408
 5 3rd   -0.528
 6 3rd    0.415
 7 1st   -0.142
 8 Crew  -0.286
 9 1st    0.114
10 1st   -1.34 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_17_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                  diff        lwr       upr     p adj
2nd-1st  -0.0832625535 -1.0657594 0.8992343 0.9958776
3rd-1st  -0.4965960818 -1.4790929 0.4859007 0.5382667
Crew-1st  0.0006832311 -0.9460744 0.9474408 1.0000000
3rd-2nd  -0.4133335283 -1.3958303 0.5691633 0.6783714
Crew-2nd  0.0839457846 -0.8628118 1.0307034 0.9952876
Crew-3rd  0.4972793128 -0.4494783 1.4440369 0.5057813
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 Crew   0.610 
 2 Crew   0.110 
 3 1st   -1.97  
 4 1st    0.0303
 5 3rd   -0.793 
 6 2nd   -1.08  
 7 Crew  -0.413 
 8 1st   -1.92  
 9 Crew  -0.307 
10 3rd   -0.426 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_18_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
               diff        lwr       upr     p adj
2nd-1st   0.1580195 -0.7723181 1.0883571 0.9687642
3rd-1st  -0.3578741 -1.3885239 0.6727757 0.7913457
Crew-1st  0.3603245 -0.4811975 1.2018465 0.6661884
3rd-2nd  -0.5158935 -1.5076368 0.4758497 0.5140344
Crew-2nd  0.2023051 -0.5910895 0.9956996 0.9042889
Crew-3rd  0.7181986 -0.1907491 1.6271463 0.1664027
```
]

---
count: false

```
# A tibble: 50 x 2
   sex    height
   <chr>   <dbl>
 1 2nd   -0.768 
 2 1st   -0.901 
 3 Crew  -0.449 
 4 3rd   -0.306 
 5 3rd   -1.12  
 6 2nd   -0.625 
 7 3rd    0.0779
 8 Crew   0.0111
 9 2nd   -0.747 
10 2nd   -0.825 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_19_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr       upr     p adj
2nd-1st  -0.47334284 -1.3710309 0.4243452 0.5024502
3rd-1st  -0.29575403 -1.1934421 0.6019340 0.8161022
Crew-1st -0.26589576 -1.1454468 0.6136553 0.8513662
3rd-2nd   0.17758881 -0.7378769 1.0930546 0.9545842
Crew-2nd  0.20744708 -0.6902410 1.1051351 0.9264729
Crew-3rd  0.02985827 -0.8678298 0.9275463 0.9997464
```
]

---
count: false

```
# A tibble: 50 x 2
   sex   height
   <chr>  <dbl>
 1 2nd    0.233
 2 2nd   -0.927
 3 1st    0.840
 4 1st    0.101
 5 1st   -0.175
 6 3rd   -0.687
 7 1st   -2.54 
 8 3rd   -2.06 
 9 3rd    0.858
10 2nd    0.351
# … with 40 more rows
```

![](statistical_independence_files/figure-html/c_and_d3_20_20_output-1.png)

```
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = height_data$height ~ height_data$sex)

$`height_data$sex`
                diff        lwr      upr     p adj
2nd-1st   0.27839893 -0.9749367 1.531735 0.9339733
3rd-1st   0.15182387 -1.0293735 1.333021 0.9859901
Crew-1st  0.19796508 -1.0040174 1.399948 0.9713849
3rd-2nd  -0.12657506 -1.3363374 1.083187 0.9923242
Crew-2nd -0.08043385 -1.3104988 1.149631 0.9980906
Crew-3rd  0.04614121 -1.1103348 1.202617 0.9995630
```
]

---

# Chi-squared test

`$$\chi$$`

Two-by-two

---

```r
library(tidyverse)
tibble(sex = sample(x = c("male","female"),
                  size = 100,
                  replace = T),
       handed = sample(x = c("left", "right"),
                  size = 100,
                  replace = T,
                  prob = c(.3,.7))) ->
my_data ; my_data
#
#
#
#
#
#
#
#
#
#
library(ggmosaic) # geom_mosaic() and product()  # geom_mosaic() and product()
ggplot(data = my_data) +
  geom_mosaic(aes(x = product(sex),
                  fill = handed))
#
#
#
#
#
#
#
#
#
# contingency table or two by two table
table(x = my_data$sex, y = my_data$handed)
#
#
#
#
#
chisq.test(x = my_data$sex,
           y = my_data$handed)
```
]
 
.panel2-d_and_d-20[

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female right 
 2 female left  
 3 male   right 
 4 male   right 
 5 female right 
 6 female right 
 7 male   right 
 8 male   right 
 9 female right 
10 female left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_01_output-1.png)

```
        y
x        left right
  female   16    35
  male     13    36
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.09797, df = 1, p-value = 0.7543
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   right 
 2 male   right 
 3 female right 
 4 male   right 
 5 male   right 
 6 male   right 
 7 male   right 
 8 male   left  
 9 male   left  
10 female right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_02_output-1.png)

```
        y
x        left right
  female   12    31
  male     24    33
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 1.5726, df = 1, p-value = 0.2098
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female right 
 2 female left  
 3 male   right 
 4 male   right 
 5 male   right 
 6 male   right 
 7 male   right 
 8 male   left  
 9 male   right 
10 male   left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_03_output-1.png)

```
        y
x        left right
  female   16    39
  male     21    24
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 2.5692, df = 1, p-value = 0.109
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   left  
 2 male   left  
 3 female right 
 4 female right 
 5 female right 
 6 male   left  
 7 male   right 
 8 male   right 
 9 male   left  
10 female left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_04_output-1.png)

```
        y
x        left right
  female   13    40
  male     15    32
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.35756, df = 1, p-value = 0.5499
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female right 
 2 female right 
 3 male   right 
 4 female right 
 5 male   right 
 6 male   right 
 7 male   right 
 8 female left  
 9 male   right 
10 male   right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_05_output-1.png)

```
        y
x        left right
  female    8    43
  male     13    36
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 1.1781, df = 1, p-value = 0.2777
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female left  
 2 female left  
 3 female right 
 4 male   left  
 5 female right 
 6 female right 
 7 male   right 
 8 male   left  
 9 male   right 
10 female left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_06_output-1.png)

```
        y
x        left right
  female   18    33
  male     15    34
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.081245, df = 1, p-value = 0.7756
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   right 
 2 female right 
 3 male   right 
 4 female right 
 5 female left  
 6 male   right 
 7 male   left  
 8 female left  
 9 male   right 
10 female right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_07_output-1.png)

```
        y
x        left right
  female   10    34
  male     16    40
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.18638, df = 1, p-value = 0.6659
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female left  
 2 female right 
 3 female left  
 4 female right 
 5 female left  
 6 male   left  
 7 female right 
 8 female right 
 9 female left  
10 female left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_08_output-1.png)

```
        y
x        left right
  female   21    33
  male     13    33
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.82159, df = 1, p-value = 0.3647
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   left  
 2 male   right 
 3 female left  
 4 male   left  
 5 male   right 
 6 female right 
 7 female right 
 8 female left  
 9 male   right 
10 male   right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_09_output-1.png)

```
        y
x        left right
  female   22    33
  male     14    31
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.5068, df = 1, p-value = 0.4765
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female right 
 2 female right 
 3 female right 
 4 male   right 
 5 male   left  
 6 female left  
 7 male   right 
 8 female right 
 9 female left  
10 female right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_10_output-1.png)

```
        y
x        left right
  female   16    35
  male     20    29
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.60087, df = 1, p-value = 0.4382
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   right 
 2 male   left  
 3 male   left  
 4 female left  
 5 male   right 
 6 female left  
 7 male   right 
 8 female right 
 9 female left  
10 male   right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_11_output-1.png)

```
        y
x        left right
  female   19    35
  male     18    28
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.039791, df = 1, p-value = 0.8419
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female right 
 2 female right 
 3 female right 
 4 female left  
 5 female right 
 6 female right 
 7 female right 
 8 female right 
 9 female right 
10 female left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_12_output-1.png)

```
        y
x        left right
  female   14    46
  male     13    27
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.61094, df = 1, p-value = 0.4344
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   right 
 2 male   right 
 3 male   right 
 4 male   right 
 5 female left  
 6 male   left  
 7 male   right 
 8 male   right 
 9 female right 
10 female left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_13_output-1.png)

```
        y
x        left right
  female   15    34
  male     15    36
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 1.6836e-30, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   right 
 2 female right 
 3 female right 
 4 female right 
 5 female right 
 6 female left  
 7 male   left  
 8 male   right 
 9 male   right 
10 female right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_14_output-1.png)

```
        y
x        left right
  female   19    35
  male     14    32
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.084193, df = 1, p-value = 0.7717
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   right 
 2 female left  
 3 male   right 
 4 male   right 
 5 male   left  
 6 female right 
 7 male   right 
 8 female right 
 9 female left  
10 male   right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_15_output-1.png)

```
        y
x        left right
  female   11    35
  male     16    38
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.17288, df = 1, p-value = 0.6776
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female right 
 2 female right 
 3 female right 
 4 female left  
 5 male   left  
 6 male   right 
 7 female right 
 8 male   right 
 9 male   left  
10 female right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_16_output-1.png)

```
        y
x        left right
  female   13    44
  male     14    29
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.73942, df = 1, p-value = 0.3898
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   left  
 2 female right 
 3 male   left  
 4 female right 
 5 male   right 
 6 male   left  
 7 female right 
 8 male   right 
 9 female right 
10 female left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_17_output-1.png)

```
        y
x        left right
  female    9    38
  male     22    31
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 4.8243, df = 1, p-value = 0.02806
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 male   right 
 2 female right 
 3 male   left  
 4 male   right 
 5 male   right 
 6 male   right 
 7 female right 
 8 female right 
 9 male   right 
10 female right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_18_output-1.png)

```
        y
x        left right
  female    8    33
  male     17    42
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.67521, df = 1, p-value = 0.4112
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female left  
 2 female left  
 3 male   left  
 4 male   right 
 5 male   right 
 6 male   right 
 7 female left  
 8 male   right 
 9 male   left  
10 male   left  
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_19_output-1.png)

```
        y
x        left right
  female   20    29
  male     11    40
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 3.4752, df = 1, p-value = 0.0623
```
]

---
count: false

```
# A tibble: 100 x 2
   sex    handed
   <chr>  <chr> 
 1 female right 
 2 male   right 
 3 female right 
 4 male   right 
 5 male   right 
 6 female right 
 7 female right 
 8 male   right 
 9 female left  
10 female right 
# … with 90 more rows
```

![](statistical_independence_files/figure-html/d_and_d_20_20_output-1.png)

```
        y
x        left right
  female   13    41
  male      9    37
```

```

Pearson's Chi-squared test with Yates' continuity correction

data:  my_data$sex and my_data$handed
X-squared = 0.090181, df = 1, p-value = 0.7639
```
]

---

```r
library(tidyverse)
tibble(truth = sample(rep(c("tea", "milk"), 4))) %>%
  mutate(guess = sample(rep(c("tea", "milk"), 4))) ->
tea_data ; tea_data
#
#
#
#
#
#
#
#
#
#
#
#
#
#
library(ggmosaic) # geom_mosaic() and product()  # geom_mosaic() and product()
ggplot(data = tea_data) +
  geom_mosaic(aes(x = product(truth),
                  fill = guess))
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
chisq.test(x = tea_data$truth,
           y = tea_data$guess)
```
]
 
.panel2-lady_tea-20[

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  milk 
2 milk  milk 
3 milk  tea  
4 tea   tea  
5 milk  tea  
6 tea   tea  
7 tea   milk 
8 tea   milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_01_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   milk 
2 tea   tea  
3 milk  milk 
4 milk  milk 
5 milk  tea  
6 milk  milk 
7 tea   tea  
8 tea   tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_02_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   tea  
2 tea   milk 
3 tea   milk 
4 milk  milk 
5 tea   tea  
6 milk  tea  
7 milk  tea  
8 milk  milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_03_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  tea  
2 tea   tea  
3 milk  milk 
4 tea   milk 
5 milk  tea  
6 tea   milk 
7 tea   tea  
8 milk  milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_04_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  milk 
2 milk  tea  
3 milk  milk 
4 tea   milk 
5 tea   tea  
6 tea   milk 
7 tea   tea  
8 milk  tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_05_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  milk 
2 milk  tea  
3 tea   milk 
4 milk  tea  
5 tea   tea  
6 tea   milk 
7 milk  milk 
8 tea   tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_06_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   tea  
2 milk  tea  
3 tea   tea  
4 tea   milk 
5 milk  milk 
6 milk  milk 
7 tea   tea  
8 milk  milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_07_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   tea  
2 milk  tea  
3 milk  tea  
4 tea   milk 
5 milk  milk 
6 tea   milk 
7 milk  tea  
8 tea   milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_08_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   tea  
2 milk  milk 
3 milk  tea  
4 tea   milk 
5 tea   tea  
6 tea   tea  
7 milk  milk 
8 milk  milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_09_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  milk 
2 milk  tea  
3 tea   tea  
4 milk  milk 
5 milk  milk 
6 tea   tea  
7 tea   tea  
8 tea   milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_10_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   milk 
2 milk  tea  
3 tea   tea  
4 milk  milk 
5 milk  tea  
6 milk  milk 
7 tea   milk 
8 tea   tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_11_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   tea  
2 tea   milk 
3 milk  milk 
4 milk  tea  
5 milk  milk 
6 milk  milk 
7 tea   tea  
8 tea   tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_12_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  tea  
2 tea   tea  
3 tea   milk 
4 tea   tea  
5 milk  milk 
6 milk  milk 
7 tea   tea  
8 milk  milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_13_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  tea  
2 tea   milk 
3 milk  tea  
4 tea   tea  
5 milk  milk 
6 milk  tea  
7 tea   milk 
8 tea   milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_14_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  tea  
2 milk  milk 
3 milk  tea  
4 tea   milk 
5 tea   milk 
6 milk  milk 
7 tea   tea  
8 tea   tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_15_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  milk 
2 tea   milk 
3 milk  milk 
4 tea   milk 
5 tea   tea  
6 milk  tea  
7 milk  tea  
8 tea   tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_16_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  tea  
2 tea   tea  
3 tea   milk 
4 tea   tea  
5 tea   milk 
6 milk  milk 
7 milk  tea  
8 milk  milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_17_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   milk 
2 milk  tea  
3 tea   tea  
4 milk  milk 
5 tea   milk 
6 milk  milk 
7 milk  tea  
8 tea   tea  
```

![](statistical_independence_files/figure-html/lady_tea_20_18_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 milk  tea  
2 tea   milk 
3 tea   milk 
4 milk  tea  
5 tea   tea  
6 tea   milk 
7 milk  tea  
8 milk  milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_19_output-1.png)

```

Pearson's Chi-squared test with Yates' continuity correction

data:  tea_data$truth and tea_data$guess
X-squared = 0.5, df = 1, p-value = 0.4795
```
]

---
count: false

```
# A tibble: 8 x 2
  truth guess
  <chr> <chr>
1 tea   milk 
2 milk  tea  
3 milk  milk 
4 tea   tea  
5 milk  milk 
6 tea   tea  
7 milk  tea  
8 tea   milk 
```

![](statistical_independence_files/figure-html/lady_tea_20_20_output-1.png)

```

Pearson's Chi-squared test

data:  tea_data$truth and tea_data$guess
X-squared = 0, df = 1, p-value = 1
```
]

---

```r
library(tidyverse)
tibble(class = sample(x = c("1st","2nd","3rd","Crew"),
                  size = 50,
                  replace = T)) %>%
  mutate(survive = sample(x = c("Survive", "Perish"),
                          size = 50,
                          replace = T,
                          prob = c(.3, .7))) ->
fake_titanic ; fake_titanic
#
#
#
#
#
#
#
#
#
#
library(ggmosaic) # geom_mosaic() and product()  # geom_mosaic() and product()
ggplot(data = fake_titanic) +
  geom_mosaic(aes(x = product(class),
                  fill = survive)) -> g1
ggplot(data = fake_titanic) +
  aes(x = class) +
  aes(y = survive) +
  geom_count(alpha = .5, color = "blue") +
  geom_jitter(width = .2, height = .2) -> g2
library(patchwork)
g1 + g2
#
#
#
#
#
#
#
#
#
#
#
#
#
chisq.test(x = fake_titanic$class,
           y = fake_titanic$survive)
```
]
 
.panel2-d4_d2-20[

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 2nd   Survive
 2 1st   Survive
 3 2nd   Perish 
 4 Crew  Perish 
 5 1st   Perish 
 6 2nd   Survive
 7 1st   Perish 
 8 3rd   Survive
 9 2nd   Perish 
10 Crew  Survive
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_01_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 2.3518, df = 3, p-value = 0.5027
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 3rd   Survive
 2 Crew  Perish 
 3 Crew  Perish 
 4 3rd   Perish 
 5 2nd   Perish 
 6 Crew  Survive
 7 3rd   Perish 
 8 3rd   Perish 
 9 2nd   Perish 
10 3rd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_02_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 12.751, df = 3, p-value = 0.005207
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 2nd   Perish 
 2 2nd   Perish 
 3 2nd   Survive
 4 3rd   Survive
 5 Crew  Survive
 6 Crew  Perish 
 7 3rd   Survive
 8 Crew  Survive
 9 Crew  Perish 
10 3rd   Survive
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_03_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 8.9383, df = 3, p-value = 0.03012
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 2nd   Perish 
 2 3rd   Perish 
 3 1st   Perish 
 4 2nd   Survive
 5 Crew  Perish 
 6 3rd   Perish 
 7 2nd   Perish 
 8 Crew  Perish 
 9 2nd   Perish 
10 1st   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_04_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 3.1174, df = 3, p-value = 0.3739
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 2nd   Survive
 2 Crew  Survive
 3 2nd   Perish 
 4 Crew  Perish 
 5 2nd   Perish 
 6 1st   Perish 
 7 2nd   Perish 
 8 3rd   Perish 
 9 3rd   Survive
10 3rd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_05_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 2.4046, df = 3, p-value = 0.4928
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 2nd   Perish 
 2 Crew  Perish 
 3 3rd   Perish 
 4 3rd   Perish 
 5 1st   Survive
 6 2nd   Perish 
 7 3rd   Perish 
 8 Crew  Perish 
 9 3rd   Survive
10 Crew  Survive
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_06_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 0.42067, df = 3, p-value = 0.9359
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 3rd   Perish 
 2 Crew  Perish 
 3 Crew  Perish 
 4 2nd   Perish 
 5 2nd   Perish 
 6 3rd   Perish 
 7 Crew  Perish 
 8 1st   Perish 
 9 3rd   Perish 
10 3rd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_07_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 0.47864, df = 3, p-value = 0.9236
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 3rd   Perish 
 2 Crew  Perish 
 3 1st   Perish 
 4 3rd   Perish 
 5 2nd   Perish 
 6 2nd   Survive
 7 3rd   Perish 
 8 3rd   Survive
 9 3rd   Perish 
10 3rd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_08_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 0.50755, df = 3, p-value = 0.9172
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 Crew  Perish 
 2 1st   Survive
 3 2nd   Survive
 4 Crew  Perish 
 5 3rd   Perish 
 6 3rd   Perish 
 7 3rd   Survive
 8 Crew  Perish 
 9 3rd   Perish 
10 2nd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_09_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 1.7228, df = 3, p-value = 0.6319
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 2nd   Survive
 2 Crew  Perish 
 3 2nd   Perish 
 4 3rd   Perish 
 5 2nd   Perish 
 6 2nd   Survive
 7 3rd   Perish 
 8 Crew  Perish 
 9 3rd   Perish 
10 3rd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_10_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 0.94684, df = 3, p-value = 0.8141
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 Crew  Survive
 2 3rd   Perish 
 3 Crew  Perish 
 4 3rd   Survive
 5 1st   Survive
 6 2nd   Survive
 7 Crew  Perish 
 8 1st   Perish 
 9 3rd   Perish 
10 1st   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_11_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 1.976, df = 3, p-value = 0.5774
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 1st   Perish 
 2 2nd   Perish 
 3 3rd   Survive
 4 3rd   Survive
 5 1st   Survive
 6 2nd   Survive
 7 Crew  Survive
 8 3rd   Perish 
 9 1st   Perish 
10 Crew  Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_12_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 3.8352, df = 3, p-value = 0.2798
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 Crew  Perish 
 2 3rd   Perish 
 3 3rd   Survive
 4 2nd   Perish 
 5 1st   Survive
 6 2nd   Perish 
 7 Crew  Perish 
 8 2nd   Perish 
 9 Crew  Survive
10 1st   Survive
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_13_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 1.295, df = 3, p-value = 0.7303
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 3rd   Perish 
 2 1st   Perish 
 3 1st   Perish 
 4 1st   Perish 
 5 3rd   Perish 
 6 1st   Perish 
 7 3rd   Survive
 8 1st   Perish 
 9 3rd   Survive
10 Crew  Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_14_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 1.6422, df = 3, p-value = 0.6498
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 3rd   Survive
 2 1st   Perish 
 3 1st   Perish 
 4 2nd   Perish 
 5 Crew  Perish 
 6 3rd   Perish 
 7 2nd   Survive
 8 3rd   Perish 
 9 Crew  Perish 
10 2nd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_15_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 2.4201, df = 3, p-value = 0.4899
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 Crew  Perish 
 2 3rd   Survive
 3 Crew  Perish 
 4 2nd   Perish 
 5 1st   Survive
 6 Crew  Perish 
 7 2nd   Perish 
 8 1st   Survive
 9 Crew  Perish 
10 3rd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_16_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 5.7481, df = 3, p-value = 0.1245
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 Crew  Perish 
 2 2nd   Survive
 3 Crew  Perish 
 4 Crew  Perish 
 5 Crew  Perish 
 6 1st   Survive
 7 2nd   Perish 
 8 Crew  Survive
 9 2nd   Perish 
10 3rd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_17_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 2.5353, df = 3, p-value = 0.4689
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 3rd   Perish 
 2 1st   Survive
 3 3rd   Perish 
 4 1st   Perish 
 5 2nd   Perish 
 6 Crew  Perish 
 7 1st   Perish 
 8 3rd   Survive
 9 1st   Perish 
10 1st   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_18_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 2.4223, df = 3, p-value = 0.4895
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 2nd   Perish 
 2 1st   Survive
 3 2nd   Perish 
 4 3rd   Perish 
 5 1st   Perish 
 6 Crew  Perish 
 7 1st   Perish 
 8 3rd   Perish 
 9 3rd   Survive
10 2nd   Perish 
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_19_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 0.4343, df = 3, p-value = 0.9331
```
]

---
count: false

```
# A tibble: 50 x 2
   class survive
   <chr> <chr>  
 1 3rd   Perish 
 2 2nd   Perish 
 3 Crew  Perish 
 4 1st   Perish 
 5 3rd   Perish 
 6 3rd   Survive
 7 1st   Survive
 8 Crew  Perish 
 9 3rd   Survive
10 Crew  Survive
# … with 40 more rows
```

![](statistical_independence_files/figure-html/d4_d2_20_20_output-1.png)

```

Pearson's Chi-squared test

data:  fake_titanic$class and fake_titanic$survive
X-squared = 2.4098, df = 3, p-value = 0.4918
```
]