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.

H₀:₁<₂;H₁:₁=₂

H₀:₁=₂;H₁:₁<₂

H₀:₁=₂;H₁:₁>₂

H₀:₁=₂;H₁:₁₂

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

The Student'st. We assume that both population distributions are approximately normal with known standard deviations.

The Student'st. We assume that both population distributions are approximately normal with unknown standard deviations.

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown 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?

At the= 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

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

At the= 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

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.

Fail to reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Reject the null hypothesis, there is insufficient evidence that the mean REM sleep time for children is more than for adults.

Reject the null hypothesis, there is sufficient evidence that the mean REM sleep time for children is more than for adults.

Fail to 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=

Explain the meaning of the confidence interval in the context of the problem.

• 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.
• 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.
• We can not make any conclusions using this confidence interval.
• 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.