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 = 10 children (9 years old) showed that they had an average REM sleep time of x1 = 2.5 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 x2 = 1.70 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?

(b) State the null and alternate hypotheses. (out of the following)

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂    

H₀: μ₁ = μ₂; H₁: μ₁ > μ₂

H₀: μ₁ < μ₂; H₁: μ₁ = μ₂

(c) What sampling distribution will you use? What assumptions are you making? (out of the following)

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

The Student's t. 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.

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

(d) What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to two decimal places.)

(e) Find (or estimate) the P-value. (Round your answer to four decimal places.)

(f) Sketch the sampling distribution and show the area corresponding to the P-value.

(g) Based on your answers in parts (a) to (e), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (out of the following)

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

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

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

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

(h) Interpret your conclusion in the context of the application. (out of the following)

Fail to reject the null hypothesis, there is sufficient 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.    

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 sufficient evidence that the mean REM sleep time for children is more than for adults.

Answer 1

a)

0.01 is the levl of significance

b/0

H0: μ1 = μ2; H1: μ1 > μ2

c)

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

d)

Pooled Variance
sp = sqrt(s1^2/n1 + s2^2/n2)
sp = sqrt(0.25/10 + 0.81/10)
sp = 0.3256

Test statistic,
z = (x1bar - x2bar)/sp
z = (2.5 - 1.7)/0.3256
z = 2.46

e)
P-value Approach
P-value = 0.0069
As P-value < 0.01, reject the null hypothesis.

g)

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

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