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 = 9 children (9 years old) showed that they had an average REM sleep time of x₁ = 2.6 hours per night. From previous studies, it is known that σ₁ = 0.7 hour. Another random sample of n₂ = 9 adults showed that they had an average REM sleep time of x₂ = 2.10 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 10% level of significance. Solve the problem using both the traditional method and the P-value method. (Test the difference μ₁ − μ₂. Round the test statistic and critical value to two decimal places. Round the P-value to four decimal places.)

test statistic
critical value
P-value

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ1 = μ2
Alternative Hypothesis, Ha: μ1 > μ2

Pooled Variance
sp = sqrt(s1^2/n1 + s2^2/n2)
sp = sqrt(0.49/9 + 0.81/9)
sp = 0.3801

Test statistic,
z = (x1bar - x2bar)/sp
z = (2.6 - 2.1)/0.3801
z = 1.32

Rejection Region
This is right tailed test, for α = 0.1
Critical value of z is 1.28.
Hence reject H0 if z > 1.28

P-value Approach
P-value = 0.0934
As P-value < 0.1, reject the null hypothesis.