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 adultst. Assume that REM sleep time is normally distributed for both children and adults. A random sample of n,- 8 children (9 years old) showed that they had an average REM sleep time of - 2.9 hours per night. From previous studies, it is known that a, - 0.6 hour. Another random sample of n - 8 adults showed that they had an average REM sleep time of x2- 2.00 hours per night. Previous studies show that z - 0.8 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. (b) What sampling distribution will you use? What assumptions are you making? O The standard normal. We assume that both population distributions are approximately normal with known standard deviations. O The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's t. We assume that both population distributions are approximately normal with known standard deviations. What is the value of the sample test statistic? (Test the difference , - H2. Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. O-3 1 -2 -1 O-3 -2 -1 1 2 -2 -1 1 2 O-3 -2 -1 (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? O At the a - 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. O At the a - 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. O At the a = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. O At the a - 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. O Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. O Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults. O Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults. O Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.