Hypothesis Testing

Answers to questions from Quiz 13.

Load Libraries

library(tidyverse)
library(infer)
library(skimr)
library(here)

1

• Set random seed generator to 123

set.seed(123)

• hr_2_tidy.csv is the name of your data subset
• Read it into and assign to hr

hr  <- read_csv("https://estanny.com/static/week13/data/hr_2_tidy.csv",
                col_types = "fddfff")

• use the skim to summarize the data in hr

skim(hr)

Table 1: Data summary

Name	hr
Number of rows	500
Number of columns	6
_______________________
Column type frequency:
factor	4
numeric	2
________________________
Group variables	None

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
gender	0	1	FALSE	2	mal: 256, fem: 244
evaluation	0	1	FALSE	4	bad: 154, fai: 142, goo: 108, ver: 96
salary	0	1	FALSE	6	lev: 95, lev: 94, lev: 87, lev: 85
status	0	1	FALSE	3	fir: 194, pro: 179, ok: 127

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
age	0	1	39.86	11.55	20.3	29.60	40.2	50.1	59.9	▇▇▇▇▇
hours	0	1	49.39	13.15	35.0	37.48	45.6	58.9	79.9	▇▃▂▂▂

• specify that hours is the variable of interest

hr %>% 
specify(response = hours)

Response: hours (numeric)
# A tibble: 500 x 1
   hours
   <dbl>
 1  78.1
 2  35.1
 3  36.9
 4  38.5
 5  36.1
 6  78.1
 7  76  
 8  35.6
 9  35.6
10  56.8
# ... with 490 more rows

• hypothesize that the average hours worked is 48

hr %>% 
  specify(response = hours)  %>% 
  hypothesize(null = "point", mu = 48)

Response: hours (numeric)
Null Hypothesis: point
# A tibble: 500 x 1
   hours
   <dbl>
 1  78.1
 2  35.1
 3  36.9
 4  38.5
 5  36.1
 6  78.1
 7  76  
 8  35.6
 9  35.6
10  56.8
# ... with 490 more rows

• generate 1000 replicates representing the null hypothesis

hr %>% 
  specify(response = hours)  %>% 
  hypothesize(null = "point", mu = 48)  %>% 
  generate(reps = 1000, type = "bootstrap")

Response: hours (numeric)
Null Hypothesis: point
# A tibble: 500,000 x 2
# Groups:   replicate [1,000]
   replicate hours
       <int> <dbl>
 1         1  39.7
 2         1  44.3
 3         1  46.8
 4         1  33.7
 5         1  39.6
 6         1  39.5
 7         1  40.5
 8         1  55.8
 9         1  72.6
10         1  35.7
# ... with 499,990 more rows

• calculate the distribution of statistics from the generated data
• Assign the output null_t_distribution
• Display null_t_distribution

null_t_distribution <- hr  %>% 
  specify(response = age)  %>% 
  hypothesize(null = "point", mu = 48)  %>% 
  generate(reps = 1000, type = "bootstrap")  %>%
  calculate(stat = "t")
  
  null_t_distribution

# A tibble: 1,000 x 2
   replicate     stat
       <int>    <dbl>
 1         1  0.144  
 2         2 -1.72   
 3         3  0.404  
 4         4 -1.11   
 5         5  0.00894
 6         6  1.46   
 7         7 -0.905  
 8         8 -0.663  
 9         9  0.291  
10        10  3.09   
# ... with 990 more rows

• visualize the simulated null distribution

visualize(null_t_distribution)

• calculate the statistic from your observed data
• Assign the output observed_t_statistic
• Display observed_t_statistic

observed_t_statistic <- hr  %>%
  specify(response = hours)  %>% 
  hypothesize(null = "point", mu = 48)  %>%
  calculate(stat = "t")

observed_t_statistic

# A tibble: 1 x 1
   stat
  <dbl>
1  2.37

• get_p_value from the simulated null distribution and the observed statistic

null_t_distribution  %>% 
  get_p_value(obs_stat = observed_t_statistic, direction = "two-sided")

# A tibble: 1 x 1
  p_value
    <dbl>
1   0.014

• shade_p_value on the simulated null distribution

null_t_distribution  %>% 
  visualize() +
  shade_p_value(obs_stat = observed_t_statistic, direction = "two-sided")

2

• hr_3_tidy.csv is the name of your data subset
• Read it into and assign to hr_2

hr_2 <- read_csv("https://estanny.com/static/week13/data/hr_3_tidy.csv",
                  col_types = "fddfff") 

• Q: Is the average number of hours worked the same for both genders in hr_2?
• use skim to summarize the data in hr_2 by gender

hr_2 %>% 
  group_by(gender) %>% 
  skim()

Table 2: Data summary

Name	Piped data
Number of rows	500
Number of columns	6
_______________________
Column type frequency:
factor	3
numeric	2
________________________
Group variables	gender

Variable type: factor

skim_variable	gender	n_missing	complete_rate	ordered	n_unique	top_counts
evaluation	male	0	1	FALSE	4	bad: 72, fai: 67, goo: 61, ver: 47
evaluation	female	0	1	FALSE	4	bad: 76, fai: 71, goo: 61, ver: 45
salary	male	0	1	FALSE	6	lev: 47, lev: 43, lev: 43, lev: 42
salary	female	0	1	FALSE	6	lev: 51, lev: 46, lev: 45, lev: 43
status	male	0	1	FALSE	3	fir: 98, pro: 81, ok: 68
status	female	0	1	FALSE	3	fir: 98, pro: 91, ok: 64

Variable type: numeric

skim_variable	gender	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
age	male	0	1	38.23	10.86	20	28.9	37.9	47.05	59.9	▇▇▇▇▅
age	female	0	1	40.56	11.67	20	31.0	40.3	50.50	59.8	▆▆▇▆▇
hours	male	0	1	49.55	13.11	35	38.4	45.4	57.65	79.9	▇▃▂▂▂
hours	female	0	1	49.80	13.38	35	38.2	45.6	59.40	79.8	▇▂▃▂▂

• Use geom_boxplot to plot distributions of hours worked by gender

hr_2 %>% 
  ggplot(aes(x = gender, y = hours)) +
  geom_boxplot()

-hypothesize that the number of hours worked and gender are independent

hr_2 %>% 
  specify(response = hours, explanatory = gender)  %>% 
  hypothesize(null = "independence")

Response: hours (numeric)
Explanatory: gender (factor)
Null Hypothesis: independence
# A tibble: 500 x 2
   hours gender
   <dbl> <fct> 
 1  49.6 male  
 2  39.2 female
 3  63.2 female
 4  42.2 male  
 5  54.7 male  
 6  54.3 female
 7  37.3 female
 8  45.6 female
 9  35.1 female
10  53   male  
# ... with 490 more rows

• generate 1000 replicates representing the null hypothesis

hr_2 %>% 
  specify(response = hours, explanatory = gender)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute")

Response: hours (numeric)
Explanatory: gender (factor)
Null Hypothesis: independence
# A tibble: 500,000 x 3
# Groups:   replicate [1,000]
   hours gender replicate
   <dbl> <fct>      <int>
 1  55.7 male           1
 2  35.5 female         1
 3  35.1 female         1
 4  44.2 male           1
 5  52.8 male           1
 6  46   female         1
 7  41.2 female         1
 8  52.9 female         1
 9  35.6 female         1
10  35   male           1
# ... with 499,990 more rows

• calculate the distribution of statistics from the generated data
• Assign the output null_distribution_2_sample_permute
• Display null_distribution_2_sample_permute

null_distribution_2_sample_permute  <- hr_2 %>% 
  specify(response = hours, explanatory = gender)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute")  %>% 
  calculate(stat = "t", order = c("female", "male"))

null_distribution_2_sample_permute

# A tibble: 1,000 x 2
   replicate    stat
       <int>   <dbl>
 1         1 -1.81  
 2         2 -1.29  
 3         3  0.0525
 4         4 -0.793 
 5         5  0.826 
 6         6  0.429 
 7         7  0.0843
 8         8 -0.264 
 9         9  2.42  
10        10  0.603 
# ... with 990 more rows

• visualize the simulated null distribution

visualise(null_distribution_2_sample_permute)

• calculate the statistic from your observed data
• Assign the output observed_t_2_sample_stat
• observed_t_2_sample_stat

observed_t_2_sample_stat <- hr_2 %>%
  specify(response = hours, explanatory = gender)  %>% 
  calculate(stat = "t", order = c("female", "male"))

observed_t_2_sample_stat

# A tibble: 1 x 1
   stat
  <dbl>
1 0.208

• get_p_value from the simulated null distribution and the observed statistic

null_t_distribution  %>% 
  get_p_value(obs_stat = observed_t_2_sample_stat, direction = "two-sided")

# A tibble: 1 x 1
  p_value
    <dbl>
1   0.878

• shade_p_value on the simulated null distribution

null_t_distribution  %>% 
  visualize() +
  shade_p_value(obs_stat = observed_t_2_sample_stat, direction = "two-sided")

ggsave("preview_2.png", 
       path = here("_posts", "2021-05-18-hypothesis-testing"))

3

• hr_1_tidy.csv is the name of your data subset
• Read it into and assign to hr_anova

hr_anova <- read_csv("https://estanny.com/static/week13/data/hr_1_tidy.csv", 
                     col_types = "fddfff")

• Q: Is the average number of hours worked the same for all three status (fired, ok and promoted) ?
• use skim to summarize the data in hr_anova by status

hr_anova %>% 
  group_by(status) %>% 
  skim()

Table 3: Data summary

Name	Piped data
Number of rows	500
Number of columns	6
_______________________
Column type frequency:
factor	3
numeric	2
________________________
Group variables	status

Variable type: factor

skim_variable	status	n_missing	complete_rate	ordered	n_unique	top_counts
gender	fired	0	1	FALSE	2	fem: 96, mal: 89
gender	ok	0	1	FALSE	2	fem: 77, mal: 76
gender	promoted	0	1	FALSE	2	fem: 87, mal: 75
evaluation	fired	0	1	FALSE	4	bad: 65, fai: 63, goo: 31, ver: 26
evaluation	ok	0	1	FALSE	4	bad: 69, fai: 59, goo: 15, ver: 10
evaluation	promoted	0	1	FALSE	4	ver: 63, goo: 60, fai: 20, bad: 19
salary	fired	0	1	FALSE	6	lev: 41, lev: 37, lev: 32, lev: 32
salary	ok	0	1	FALSE	6	lev: 40, lev: 37, lev: 29, lev: 23
salary	promoted	0	1	FALSE	6	lev: 37, lev: 35, lev: 29, lev: 23

Variable type: numeric

skim_variable	status	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
age	fired	0	1	38.64	11.43	20.2	28.30	38.30	47.60	59.6	▇▇▇▅▆
age	ok	0	1	41.34	12.11	20.3	31.00	42.10	51.70	59.9	▆▆▆▆▇
age	promoted	0	1	42.13	10.98	21.0	33.40	42.95	50.98	59.9	▆▅▆▇▇
hours	fired	0	1	41.67	7.88	35.0	36.10	38.90	43.90	75.5	▇▂▁▁▁
hours	ok	0	1	48.05	11.65	35.0	37.70	45.60	56.10	78.2	▇▃▃▂▁
hours	promoted	0	1	59.27	12.90	35.0	51.12	60.10	70.15	79.7	▆▅▇▇▇

• Use geom_boxplot to plot distributions of hours worked by status

hr_anova %>% 
  ggplot(aes(x = status, y = hours)) +
  geom_boxplot()

• specify the variables of interest are hours and status

hr_anova %>% 
  specify(response = hours, explanatory = status)

Response: hours (numeric)
Explanatory: status (factor)
# A tibble: 500 x 2
   hours status  
   <dbl> <fct>   
 1  36.5 fired   
 2  55.8 ok      
 3  35   fired   
 4  52   promoted
 5  35.1 ok      
 6  36.3 ok      
 7  40.1 promoted
 8  42.7 fired   
 9  66.6 promoted
10  35.5 ok      
# ... with 490 more rows

• hypothesize that the number of hours worked and status are independent

hr_anova %>% 
  specify(response = hours, explanatory = status) %>% 
  hypothesise(null = "independence")

Response: hours (numeric)
Explanatory: status (factor)
Null Hypothesis: independence
# A tibble: 500 x 2
   hours status  
   <dbl> <fct>   
 1  36.5 fired   
 2  55.8 ok      
 3  35   fired   
 4  52   promoted
 5  35.1 ok      
 6  36.3 ok      
 7  40.1 promoted
 8  42.7 fired   
 9  66.6 promoted
10  35.5 ok      
# ... with 490 more rows

• generate 1000 replicates representing the null hypothesis

hr_anova %>% 
  specify(response = hours, explanatory = status)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute")

Response: hours (numeric)
Explanatory: status (factor)
Null Hypothesis: independence
# A tibble: 500,000 x 3
# Groups:   replicate [1,000]
   hours status   replicate
   <dbl> <fct>        <int>
 1  40.3 fired            1
 2  40.3 ok               1
 3  37.3 fired            1
 4  50.5 promoted         1
 5  35.1 ok               1
 6  67.8 ok               1
 7  39.3 promoted         1
 8  35.7 fired            1
 9  40.2 promoted         1
10  38.4 ok               1
# ... with 499,990 more rows

• calculate the distribution of statistics from the generated data
• Assign the output null_distribution_anova
• Display null_distribution_anova

null_distribution_anova <- hr_anova %>% 
  specify(response = hours, explanatory = gender)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute")  %>% 
  calculate(stat = "F")

null_distribution_anova

# A tibble: 1,000 x 2
   replicate   stat
       <int>  <dbl>
 1         1 0.365 
 2         2 0.650 
 3         3 0.185 
 4         4 0.0184
 5         5 0.163 
 6         6 0.0194
 7         7 4.92  
 8         8 2.11  
 9         9 0.341 
10        10 0.855 
# ... with 990 more rows

• visualize the simulated null distribution

visualize(null_distribution_anova)

• calculate the statistic from your observed data
• Assign the output observed_f_sample_stat
• Display observed_f_sample_stat

observed_f_sample_stat <- hr_anova %>%
  specify(response = hours, explanatory = status)  %>% 
  calculate(stat = "F")

observed_f_sample_stat

# A tibble: 1 x 1
   stat
  <dbl>
1  115.

• get_p_value from the simulated null distribution and the observed statistic

null_distribution_anova  %>% 
  get_p_value(obs_stat = observed_f_sample_stat, direction = "greater")

# A tibble: 1 x 1
  p_value
    <dbl>
1       0

• shade_p_value on the simulated null distribution

null_t_distribution  %>% 
  visualize() +
  shade_p_value(obs_stat = observed_f_sample_stat, direction = "greater")