Question

REM (rapid eye movement) sleep is sleep during which most dreams occur. Each night a person...

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 = 2.5 hours per night. From previous studies, it is known that σ1 = 0.6 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 = 0.5 hour. Do these data indicate that, on average, children tend to have more REM sleep than adults? Use a 5% 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

Reject the null hypothesis, there is sufficient 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 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.

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.

We reject the null hypothesis using the traditional method, but fail to reject using the P-value method.

These two methods differ slightly.

We reject the null hypothesis using the P-value method, but fail to reject using the traditional method.

 x1            = 2.50 x2            = 2.00 n1           = 11 n2           = 11 σ1           = 0.60 σ2           = 0.50 std error σ1-2=√(σ21/n1+σ22/n2)    = 0.235 test stat z =(x1-x2-Δo)/σ1-2     = 2.12

Critical value at 5% level =1.64

p value=0.0170

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.