Task 1: The critical value for independence is 3.841. The p-value is 0. This means that the observed data is not consistent with the null hypothesis, therefore there is enough evidence to reject it.

test_stat <- france_data %>%
  specify(explanatory = educ.ba, # change variable name for explanatory variable
          response = sbbsntx) %>% # change variable name for outcome of interest
  hypothesize(null = "independence") %>%
  calculate(stat = "t") # replace in between quotation marks appropriate test statistic
## Warning: Removed 14528 rows containing missing values.
## Warning: The statistic is based on a difference or ratio; by default, for
## difference-based statistics, the explanatory variable is subtracted in the
## order "BA or more" - "No BA", or divided in the order "BA or more" / "No BA"
## for ratio-based statistics. To specify this order yourself, supply `order =
## c("BA or more", "No BA")` to the calculate() function.
print(test_stat$stat)
##        t 
## 4.450111
null_distribution <- france_data %>%
  specify(explanatory = educ.ba,
          response = sbbsntx) %>%
  hypothesize(null = "independence") %>%
  generate(reps = 1000, type = "permute") %>% # only line we are adding, use reps = 1000 as standard but change if too computationally demanding
  calculate(stat = "t")
## Warning: Removed 14528 rows containing missing values.
## The statistic is based on a difference or ratio; by default, for
## difference-based statistics, the explanatory variable is subtracted in the
## order "BA or more" - "No BA", or divided in the order "BA or more" / "No BA"
## for ratio-based statistics. To specify this order yourself, supply `order =
## c("BA or more", "No BA")` to the calculate() function.
p_val <- null_distribution %>% # replace name here if you assigned something other than null_distribution above
  get_pvalue(obs_stat = test_stat, direction = "two-sided") # would only replace test_stat if assigned another name in Step 1
## Warning: Please be cautious in reporting a p-value of 0. This result is an approximation
## based on the number of `reps` chosen in the `generate()` step.
## ℹ See `get_p_value()` (`?infer::get_p_value()`) for more information.
p_val
## # A tibble: 1 × 1
##   p_value
##     <dbl>
## 1       0

Task 2: considering the p-value is 0, which is less than the significant value, 0.05. This is evidence against the null hypothesis.

null_distribution %>%
  visualize() +
  shade_p_value(obs_stat = test_stat, direction = "two-sided")

null_distribution
## Response: sbbsntx (numeric)
## Explanatory: educ.ba (factor)
## Null Hypothesis: independence
## # A tibble: 1,000 × 2
##    replicate    stat
##        <int>   <dbl>
##  1         1  0.138 
##  2         2 -1.15  
##  3         3  0.699 
##  4         4  0.0849
##  5         5  1.27  
##  6         6  0.430 
##  7         7  0.804 
##  8         8 -1.67  
##  9         9 -0.609 
## 10        10  0.0619
## # ℹ 990 more rows
conf_int <- null_distribution %>%
  get_confidence_interval(level = 0.95, type = "percentile")


null_distribution %>%
  visualize() +
  shade_p_value(obs_stat = test_stat, direction = "two-sided") +
  shade_confidence_interval(endpoints = conf_int)

null_distribution
## Response: sbbsntx (numeric)
## Explanatory: educ.ba (factor)
## Null Hypothesis: independence
## # A tibble: 1,000 × 2
##    replicate    stat
##        <int>   <dbl>
##  1         1  0.138 
##  2         2 -1.15  
##  3         3  0.699 
##  4         4  0.0849
##  5         5  1.27  
##  6         6  0.430 
##  7         7  0.804 
##  8         8 -1.67  
##  9         9 -0.609 
## 10        10  0.0619
## # ℹ 990 more rows

Task 3: After visualizing with confidence intervals, the observed test statistic falls outside of the null distribution. Therefore, the null hypothesis should be rejected.

test_stat <- france_data %>%
  specify(explanatory = educ.ba, # change variable name for explanatory variable
          response = ccrdprs) %>% # change variable name for outcome of interest
  hypothesize(null = "independence") %>%
  calculate(stat = "t") # replace in between quotation marks appropriate test statistic
## Warning: Removed 14645 rows containing missing values.
## Warning: The statistic is based on a difference or ratio; by default, for
## difference-based statistics, the explanatory variable is subtracted in the
## order "BA or more" - "No BA", or divided in the order "BA or more" / "No BA"
## for ratio-based statistics. To specify this order yourself, supply `order =
## c("BA or more", "No BA")` to the calculate() function.
print(test_stat$stat)
##         t 
## -1.545298
null_distribution <- france_data %>%
  specify(explanatory = educ.ba,
          response = trstplt) %>%
  hypothesize(null = "independence") %>%
  generate(reps = 1000, type = "permute") %>% # only line we are adding, use reps = 1000 as standard but change if too computationally demanding
  calculate(stat = "t")
## Warning: The statistic is based on a difference or ratio; by default, for
## difference-based statistics, the explanatory variable is subtracted in the
## order "BA or more" - "No BA", or divided in the order "BA or more" / "No BA"
## for ratio-based statistics. To specify this order yourself, supply `order =
## c("BA or more", "No BA")` to the calculate() function.
p_val <- null_distribution %>% # replace name here if you assigned something other than null_distribution above
  get_pvalue(obs_stat = test_stat, direction = "two-sided") # would only replace test_stat if assigned another name in Step 1

p_val
## # A tibble: 1 × 1
##   p_value
##     <dbl>
## 1   0.112

Task 4: p-value is >0.05. This means that there was no significant change/effect observed. This is evidence that the null hypothesis is acceptable.

null_distribution %>%
  visualize() +
  shade_p_value(obs_stat = test_stat, direction = "two-sided")

conf_int <- null_distribution %>%
  get_confidence_interval(level = 0.95, type = "percentile")


null_distribution %>%
  visualize() +
  shade_p_value(obs_stat = test_stat, direction = "two-sided") +
  shade_confidence_interval(endpoints = conf_int)

null_distribution
## Response: trstplt (numeric)
## Explanatory: educ.ba (factor)
## Null Hypothesis: independence
## # A tibble: 1,000 × 2
##    replicate   stat
##        <int>  <dbl>
##  1         1 -2.45 
##  2         2 -0.553
##  3         3 -0.487
##  4         4  0.780
##  5         5 -0.639
##  6         6  0.143
##  7         7 -0.334
##  8         8 -0.225
##  9         9 -0.957
## 10        10 -0.444
## # ℹ 990 more rows

Task 5: The null hypothesis should be accepted.The observed test statistic falls within the null distribution of the data.