It has long been stated that the mean temperature of humans is 98.6 degrees°F. H
ID: 3321525 • Letter: I
Question
It has long been stated that the mean temperature of humans is 98.6 degrees°F. However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6 degrees°F. They measured the temperatures of 56 healthy adults 1 to 4 times daily for 3 days, obtaining 250 measurements. The sample data resulted in a sample mean of 98.4 degrees°F and a sample standard deviation of 1degrees°F.
Use the P-value approach to conduct a hypothesis test to judge whether the mean temperature of humans is less than 98.6 degrees°F at the
alpha=0.01 level of significance.
Explanation / Answer
Given:- sample mean is X¯=98.4 and the known population standard deviation is =1, and the sample size is n=250.
(1) Null and Alternative Hypotheses
The following null and alternative hypotheses need to be tested:
Ho: =98.6
Ha: <98.6
This corresponds to a left-tailed test, for which a z-test for one mean, with known population standard deviation will be used.
(2) Rejection Region
Based on the information provided, the significance level is =0.01, and the critical value for a left-tailed test is zc=2.33.
The rejection region for this left-tailed test is R={z:z<2.33}
(3) Test Statistics
The z-statistic is computed as follows:
z=[ (X¯–0) / (/n) ] = [ (98.4–98.61) / (250/1) ] = 3.162
(4) Decision about the null hypothesis
Since it is observed that z=3.162<zc=2.33, it is then concluded that the null hypothesis is rejected.
Using the P-value approach: The p-value is p=0.0008, and since p=0.0008<0.01, it is concluded that the null hypothesis is rejected.
(5) Conclusion
It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean is less than 98.6, at the 0.01 significance level.
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