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 = 2.9 hours per night. From previous studies, it is known that σ₁ = 0.6 hour. Another random sample of n₂ = 9 adults showed that they had an average REM sleep time of x₂ = 2.40 hours per night. Previous studies show that σ₂ = 0.7 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

Conclusion

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

Compare your conclusion with the conclusion obtained by using the P-value method. Are they the same?

The conclusions obtained by using both methods are the same.These two methods differ slightly.    We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.

Answer 1

x1            =	2.90	x2            =	2.40
n1           =	9	n2           =	9
σ1           =	0.60	σ2           =	0.70
std error σ1-2=√(σ21/n1+σ22/n2)    =			0.307
test stat z =(x1-x2-Δo)/σ1-2     =			1.63

for 0.1 level with right tail test , critical z=

1.28

p value : =

0.0516

Conclusion:

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

The conclusions obtained by using both methods are the same