Question

(11) 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 = 2.6 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 = 2.10 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 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=

Answer 1

Solution:

Given:

Sample 1 Children:

Sample size = n1 = 11

Sample mean =

Population Standard Deviation =

Sample 2 Adults:

Sample size = n2 = 11

Sample mean =

Population Standard Deviation =

level of significance = 5% = 0.05

We have to test if this data indicate that, on average, children tend to have more REM sleep than adults.

Thus hypothesis of the study are:

( this indicates this is right tailed test)

Part a) Test Statistic :

Part b) Critical value:

level of significance = 5% = 0.05

For right tailed test , find area = 1 - 0.05 = 0.9500

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus Z_critical = 1.645 = 1.65

That is Critical value = 1.65

Decision Rule: reject H0, if z test statistic value > z critical value = 1.65 ,otherwise we fail to reject H0.

Since z test statistic value = 1.80 > z critical value = 1.65 , we reject H0.

thus this data indicate that, on average, children tend to have more REM sleep than adults

Part c) P-value:

For right tailed test , P-value is given by:

P-value = P( Z > z test statistic value)

P-value = P( Z > 1.80 )

P-value = 1 - P( Z < 1.80 )

Look in z table for z = 1.8 and 0.00 and find area.

P( Z< 1.80) = 0.9641

Thus

P-value = 1 - P( Z < 1.80 )

P-value = 1 - 0.9641

P-value = 0.0359

Decision rule: Reject H0, if P-value < 0.05 level of significance , otherwise we fail to reject H0.

Since P-value < 0.05 level of significance , we reject H0.

thus this data indicate that, on average, children tend to have more REM sleep than adults