Question

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 = 11 children (9 years old) showed that they had an average REM sleep time of x₁ = 2.9 hours per night. From previous studies, it is known that σ₁ = 0.8 hour. Another random sample of n₂ = 11 adults showed that they had an average REM sleep time of x₂ = 2.00 hours per night. Previous studies show that σ₂ = 0.9 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.

a. H₀: μ₁ = μ₂; H₁: μ₁ > μ₂

b. H₀: μ₁ < μ₂; H₁: μ₁ = μ₂    

c. H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

d. H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂

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

a. The Student's t. We assume that both population distributions are approximately normal with known standard deviations.

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

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

d. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. 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.

(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 α?

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

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

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

d. At the α = 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.

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

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

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

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

Answer 1

a)

0.01 is level of significance

a. H0: μ1 = μ2; H1: μ1 > μ2

b)

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

Pooled Variance
sp = sqrt(s1^2/n1 + s2^2/n2)
sp = sqrt(0.64/11 + 0.81/11)
sp = 0.3631

Test statistic,
z = (x1bar - x2bar)/sp
z = (2.9 - 2)/0.3631
z = 2.48

c)

P-value = 0.0066

d)

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

e)

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