Quiz on Hypothesis Testing

Suppose that you wanted to investigate whether there is a gender difference in musical preference among Plymouth State University students. To investigate this question, you took a small sample of PSU students. The sample data is stored in music.csv.

Import music.csv from Moodle under Date Files, and test the hypothesis as described as below.

• Null hypothesis: gender and singer are unrelated variables.
• Alternative hypothesis: male students are more likely to prefer Imagine Dragons.

# Load packages
library(dplyr)
library(ggplot2)
library(infer)

# Import data
music <- read.csv("/resources/rstudio/Business Statistics/data/music.csv")
head(music)
##   singer    sex
## 1 Grande female
## 2 Grande female
## 3 Grande female
## 4 Grande female
## 5 Grande female
## 6 Grande female

Q1 How many male students in the sample reported to prefer Imagine Dragons?

15 out of 18 males prefer dragons

music %>%
  # Count the rows by singer and sex
  count(sex, singer)
## # A tibble: 2 x 3
##   sex    singer      n
##   <fct>  <fct>   <int>
## 1 female Grande     14
## 2 male   Dragons    15

Interpretation

• No female students prefer Imagine Dragons. That is, all female students prefer Arianne Grande.
• 15 male students prefer Imagine Dragons, while only 3 male students prefer Arianne Grande.
• The sample includes 18 male students and 14 female students.

Q2 What percentage of male students in the sample reported to prefer Imagine Dragons?

83.3 percent of male students prefer dragons.

# Find proportion of each sex who were Dragons
music %>%
  # Group by sex
  group_by(sex) %>%
  # Calculate proportion Dragons summary stat
  summarise(Dragons_prop = mean(singer == "Dragons"))
## # A tibble: 2 x 2
##   sex    Dragons_prop
##   <fct>         <dbl>
## 1 female            0
## 2 male              1

Interpretation

• No female students prefer Imagine Dragons.
• 83.3% of male students prefer Imagine Dragons.
• The difference in proportions is 0.833 (male - female).
• It means that male students are more likely to prefer Imangine Dragons than female students do by 83.3%.

Q3 What is the observed difference in the proportions between male and female (male - female) students in the sample?

Most male students prefer dragons over grande where at female students prefer grande

Q4 What does this mean?

This means that the majority of males prefer dragons and the majority of females prefer grande

Q5 There might be a few negative permuted differences. What would a negative difference mean?

This would mean that the sex of the student goes against the norm and prefers the other preformer

# Calculate the observed difference in promotion rate
diff_orig <- music %>%
  # Group by sex
  group_by(sex) %>%
  # Summarize to calculate fraction Dragons
  summarise(prop_prom = mean(singer == "Dragons")) %>%
  # Summarize to calculate difference
  summarise(stat = diff(prop_prom)) %>% 
  pull()
    
# See the result
diff_orig # male - female
## [1] 1

# Create data frame of permuted differences in promotion rates
music_perm <- music %>%
  # Specify variables: singer (response variable) and sex (explanatory variable)
  specify(singer ~ sex, success = "Dragons") %>%
  # Set null hypothesis as independence: there is no gender musicrimination
  hypothesize(null = "independence") %>%
  # Shuffle the response variable, singer, one thousand times
  generate(reps = 1000, type = "permute") %>%
  # Calculate difference in proportion, male then female
  calculate(stat = "diff in props", order = c("male", "female")) # male - female
  
music_perm
## # A tibble: 1,000 x 2
##    replicate    stat
##        <int>   <dbl>
##  1         1  0.0333
##  2         2  0.310 
##  3         3 -0.243 
##  4         4  0.0333
##  5         5  0.0333
##  6         6  0.310 
##  7         7 -0.105 
##  8         8 -0.105 
##  9         9  0.0333
## 10        10  0.0333
## # ... with 990 more rows

# Using permutation data, plot stat
ggplot(music_perm, aes(x = stat)) + 
  # Add a histogram layer
  geom_histogram(binwidth = 0.01) +
  # Using original data, add a vertical line at stat
  geom_vline(aes(xintercept = diff_orig), color = "red")

Interpretation (no need to revise)

• The distribution of null statistics (permuted differences) is centered around 0.
• For example, the tallest bar in the center of the distribution indicates that difference of approximately 0 is the most likely be seesn by chance (about 380 times of 1,000) if there were no gender difference.

Q6 What is the calculated p-value? Interpret.

My p value for my data is 0, i belive it is so small that it just registers as 0

Q7 Based on the p-value you interpreted in Q6, would you reject the null hypothesis at the standard 5% significance level and accept the alternative hypothesis that male students are more likely to prefer Imagine Dragons?

Yes i would because my p value was 0

# Calculate the p-value for the original dataset
music_perm %>%
  get_p_value(obs_stat = diff_orig, direction = "greater")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1       0

Interpretation

• A value of 0.002, for example, indicates that only a 0.2% of the permuted distribution (null statistics) is more extreme than the observed difference. In other words, it would be highly unlikely to see the observed difference by chance if there was no difference across gender. Thus, we reject the null hypothesis that gender and singer are unrelated at the significance level of 5% and conclude that men are more likely to prefer Imagine Dragons.