Part 1. Paired t-tests

A sleep disorder specialist claims that herbal medicines will increase the number of hours of REM sleep per night. The specialist randomly selects 27 patients, and records the number of hours of REM sleep per night before and after taking the herbal medicines. Below are the data, ordered by patient. Assume the data are normally distributed.

Hours of REM sleep before taking the herbal medicines:

4.6, 3.1, 3.7, 3.3, 3.4, 3.4, 3.7, 5.1, 4.5, 3.2, 4.6, 6.7, 3.0, 3.7, 4.8, 4.1, 3.3, 3.7, 4.5, 3.1, 3.3, 3.8, 3.2, 3.9, 3.9, 5.1, 3.1

Hours of REM sleep after taking the herbal medicines:

4.7, 3.9, 4.1, 4.4, 4.3, 4.1, 3.4, 4.6, 3.8, 4.9, 4.4, 5.2, 3.9, 4.7, 5.7, 3.9, 3.5, 4.3, 4.7, 4.7, 3.6, 3.2, 4.2, 4.6, 4.4, 5.3, 2.1

(1a) Write the null and alternative hypotheses given the claim and with respect to D-bar and when considering pre-post. State whether this is a right-, left-, or two-tailed test. (3 points)

(1b) What is the mean, median, standard deviation of the hours of REM sleep in patients before and after taking the herbal medicines? Round your answer to one decimal place. (6 points)

(1c) Copy and paste your R code you used to test the claim using the appropriate t-test, and your output. (4 points)

(1d) What are the values of the test statistic and the p-value from (1c)? 4 points

(1e) Using alpha = 0.05 and alpha = 0.01, state the statistical decision and interpretation. (8 points)