REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person has both REM and non-REM sleep. However, it is thought that children have more REM sleep than adults. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n1 = 10 children (9 years old) showed that they had an average REM sleep time of 1 = 2.8 hours per night. From previous studies, it is known that σ1 = 0.5 hour. Another random sample of n2 = 10 adults showed that they had an average REM sleep time of 2 = 2.1 hours per night. Previous studies show that σ2 = 0.7 hour.
(a) Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.
(b) Find a 98% confidence interval for μ1 - μ2. Explain the meaning of the confidence interval in the context of the problem.
Please provide the following information for problem part (a):
(i) What is the level of significance? State the null and alternate hypotheses.
(ii) What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic?
(iii) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.
(iv) Based on your answers in parts (i) – (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
(v) Interpret your conclusion in the context of the application.