Question 1

A study was conducted on 8 pairs on twins. In each pair:

The stress level for each study participant was recorded as a score from 0 to 100.

pair twin1 twin2
1 75.25909 57.82698
2 43.47533 100.00000
3 76.59599 80.90780
4 75.44859 34.02972
5 58.29283 23.57029
6 19.20100 49.31615
7 31.42866 49.02355
8 44.10559 45.65467
  1. You want to test if sport decreases the average stress level. State null and alternative hypotheses. What type of test is appropriate in this scenario?
  1. Restate the hypotheses in terms of \(p\), the probability to observe positive difference between stress levels of twins (twin 1 - twin 2).
  1. What would be the test statistic for this test?
  1. What would be the null distribution? Draw the null distribution.
  1. What is the observed value of test statistic?
  1. Find the p-value.
  1. What conclusion can we draw at significance level 0.1?

Question 2

A study was conducted on 50 male and 50 female first-year students at U of T.

The stress level for each study participant was recorded as a score from 0 to 100 and the summary statistics were computed.

mean(male)
## [1] 50.47862
sd(male)
## [1] 18.32589
mean(female)
## [1] 58.15811
sd(female)
## [1] 23.72332
  1. You want to test if average stress level is different for male and female students. What test will you use? State null and alternative hypotheses in terms of the male and female population averages.
  1. Well, compute degrees of freedom for this test :(
  1. Compute the \(t_{df}^{\alpha/2}\) quantile for 90% confidence interval.
  1. Compute 90% confidence interval for the difference in population means.
  1. What conclusion can we draw from the confidence interval?
  1. Now find the upper 90% CI for the difference in population means.
  1. What alternative hypothesis corresponds to this CI? What conclusion can we draw from this CI?
  1. Suppose that we know that the population variances for male and female stress levels are equal, i.e. \(\sigma^2_{male} = \sigma^2_{female}\). How can you use the summary statistics to approximate the values of the population variances?
  1. Find test statistic \(t_{obs}\) for the case when \(\sigma^2_{male} = \sigma^2_{female}\).
  1. Suppose you want to check that female students are more stressed than male students for the case when \(\sigma^2_{male} = \sigma^2_{female}\). What would be the p-value?
  1. What conclusion can you make for the hypothesis from 10 at significance level \(\alpha = 0.05\)?

Question 3

A study was conducted on 50 male and 50 female first-year students at U of T.

Each study participant was asked if they feel stressed. The following results were received:

  1. You want to test if proportions of stressed male and female students are different. What test will you use? State null and alternative hypotheses.
  1. Find the value of observed statistic.
  1. Find the p-value.
  1. Can we conclude that female students stress out more often at significance level 0.05?