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 = 0.6 hour. Another random sample of n₂ = 11adults showed that they had an average REM sleep time of x₂ = 2.20 hours per night. Previous studies show that σ₂ = 0.7 hour.

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

0.01

State the null and alternate hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ > μ₂

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

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

What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate. (Test the difference μ₁ − μ₂. Round your answer to two decimal places.)


(iii) 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.

(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 α?

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

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

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 for

μ₁ − μ₂.

(Round your answers to two decimal places.)

lower limit
upper limit

I need assistance in answering the value of the sample test statistic, and sections (iii) and (b).

Answer 1

Part a)

H0: μ1 = μ2; H1: μ1 > μ2

part ii)

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

Test Statistic :-


Z = 2.52

Part iii)

P value = P ( Z < 2.5182 ) = 0.0059

Part iv)

Reject null hypothesis if P value < α = 0.01 level of significance
Since 0.0059 < 0.01 ,hence we reject null hypothesis
Result :- Reject null hypothesis

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

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

Part b)

Confidence interval :-

Z(α/2) = Z (0.02 /2) = 2.326

Lower Limit =
Lower Limit = 0.05
Upper Limit =
Upper Limit = 1.35
98% Confidence interval is ( 0.05 , 1.35 )