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 ofn₁=9children (9 years old) showed that they had an average REM sleep time ofx₁=2.9hours per night. From previous studies, it is known that₁=0.5hour. Another random sample ofn₂=9adults showed that they had an average REM sleep time ofx₂=2.2hours per night. Previous studies show that₂= 0.7hour.

(a)Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 1% level of significance.

• (i)What is the level of significance?

• State the null and alternate hypotheses.
• 1. H₀:₁=₂;H₁:₁<₂
• 2. H₀:₁=₂;H₁:₁>₂
• 3. H₀:₁=₂;H₁:₁₂
• 4. H₀:₁<₂;H₁:₁=₂

(ii)What sampling distribution will you use? What assumptions are you making?

• 1. The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
• 2. The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
• 3. The Student'st. We assume that both population distributions are approximately normal with unknown standard deviations.
• 4. The Student'st. We assume that both population distributions are approximately normal with known standard deviations.

What is the value of the sample test statistic? Compute the correspondingzort-value as appropriate. (Test the difference₁₂.Round your answer to two decimal places.)

(iii)Find (or estimate) theP-value. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to theP-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?

• 1. At the= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
• 2. At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
• 3. At the= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.
• 4. At the= 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(v) Interpret your conclusion in the context of the application.

• 1. Fail to reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.
• 2. Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
• 3. Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.
• 4. Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

(b)Find a 98% confidence interval for₁₂.

(Round your answers to two decimal places.)

• lower limit=
• upper limit=

(c) Explain the meaning of the confidence interval in the context of the problem.

• 1. Because the interval contains only positive numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is greater than that for adults.
• 2. Because the interval contains both positive and negative numbers, this indicates that at the 98% confidence level, we can not say that the population mean REM sleep time for children is greater than that for adults.
• 3. We can not make any conclusions using this confidence interval.
• 4. Because the interval contains only negative numbers, this indicates that at the 98% confidence level, the population mean REM sleep time for children is less than that for adults.