19. in a sleep study, 115 randomly selected people were asked how long they slep

Question

19. in a sleep study, 115 randomly selected people were asked how long they slept at night. From this sample of 115 people, the mean time was found to be 7.1 hours, with a standard deviation of 0.78 hours. Additionally, 24 of the 15 people surveyed said that they slept for at least 8 hours per night. Suppose the approximation method (the one involving the Central Limit Theorem) will be used to construct intervals for the relevant population mean and population proportion To construct a confidence interval for the relevant population mean based on this sample, the critical value should come from a freedom. (If the concept of degrees of freedom is not applicable, write "NA" in the second blank.) [standard normal/ (Student's) t] distribution with ILdegrees of To construct a confidence interval for the relevant population proportion based on this sample, the critical value should come from a tandard nmal Istandard normal / (Student's) t] distribution with_NA degrees of freedom. (If the concept of degrees of freedom is not applicable, write "NA" in the second blank.)

Explanation / Answer

For any sample survey/statistical test it is important to understand the assumptions that underlie performance of any test.

To start with , the data given here is a SAMPLE to study the average sleep time of 115 randomly selected people out of the entire POPULATION. The data given here is the mean of that sample and the standard deviation of that sample .

To conduct both either the z-test ( The test involving normal distribution) or the t-test ,the first and the foremost assumption that is made is that the population must be NORMALLY DISTRIBUTED meaning that it should follow a normal distribution.

The assumption that differentiates choosing one test over the other is that Z-test is performed when we have POPULATION VARIANCE value with us where t-test is performed when we have SAMPLE VARIANCE with us.

Given the particular question it is made very clear that the test is conducted on a sample and hence the values we have has sample mean and SAMPLE VARIANCE .Hence we conduct a t-test.

We say that the t-test has n-1 degree of freedom where n is the sample size .We subtract one degree of freedom because we estimate the sample variance ,which is a parameter.

HOPE THIS HELPS.

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