In this exercise involving paired differences, consider that it is reasonable to assume the populations being compared have approximately the same shape and that the distribution of paired differences is approximately symmetric.
A sample of 10 men was used in a study to test the effects of a relaxant on the time required to fall asleep. Suppose data for 10 subjects showing the number of minutes required to fall asleep with and without the relaxant follow.
Subject | Relaxant | |
---|---|---|
No | Yes | |
1 | 14 | 10 |
2 | 12 | 9 |
3 | 22 | 12 |
4 | 8 | 10 |
5 | 11 | 9 |
6 | 8 | 5 |
7 | 8 | 9 |
8 | 11 | 6 |
9 | 13 | 11 |
10 | 8 | 7 |
Use a 0.05 level of significance to determine whether the relaxant reduces the median time required to fall asleep.
State the null and alternative hypotheses.
Find the value of the test statistic.
T +=
Find the p-value. (Round your answer to four decimal places.)
p-value =
What is your conclusion?
Reject H0. There is not sufficient evidence to conclude that the relaxant reduces the median time required to fall asleep.
Do not reject H0. There is not sufficient evidence to conclude that the relaxant reduces the median time required to fall asleep.
Reject H0. There is sufficient evidence to conclude that the relaxant reduces the median time required to fall asleep.
Do not reject H0. There is sufficient evidence to conclude that the relaxant reduces the median time required to fall asleep.
The sum of positive ranks is:
W+=1.5+4+4+6.5+6.5+8+9+10=49.5
and the sum of negative ranks is:
W-=1.5+4=5.5
Hence, the test statistic is T=min{W+,W-}=min{49.5,5.5}=5.5.
(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
H0: Median (Difference) = 0
Ha: Median (Difference) > 0
(2) Rejection Region
The critical value for the significance level α=0.05 provided, and the type of tail specified is T∗=10, and the null hypothesis is rejected if T≤10.
(3) Decision about the null hypothesis
Since in this case T=5.5≤10, there is enough evidence to claim that the population median of differences is greater than 0, at the α=0.05 significance level.
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