A researcher is studying the craniofacial morphology of pati

A researcher is studying the craniofacial morphology of patients diagnosed with obstructive sleep apnea syndrome (OSAS) in otherwise healthy male Wyoming residents. Suppose that this researcher has access to a population of size 12,000 from which they can draw a sample. One of the variables available to the researcher was the body mass index of each subject.

*The actual sample size is 12,000 BMI numbers on an excel sheet, so I just included the first 100 because that should be all that is needed*

1) Find 95% confidence intervals for samples of size 10, 30, and 100.

2) Find 99% confidence intervals for the same samples used in problem 1.

3) Recall that the true population mean is 29.959. Which confidence intervals do not contain the true mean?

4) Suppose the researchers wish to perform a hypothesis test to determine whether the true mean is greater than 22, sing each of the samples from problem 1

5) Choose one of your 95% confidence intervals (make sure you specify which one you choose). Interpret it.

6) Using the same sample you chose in problem 5, interpret the corresponding 99% confidence interval.

7) Choose a p-value (specify which). Interpret it. Note: Interpreting the p-value is different than using it to make a conclusion. Try to tell me what the p-value means in context of the problem.

8) Use the p-values from each of the samples to make a reject/FTR decision at both the 0.01 and the 0.05 significance levels. Interpret (at least) one of these decisions in terms of the problem.

9) Which of the hypothesis tests is the most reliable? Are any of the tests completely unreliable? How do you know?

10) Recall that we know the true population mean. Did any hypothesis test(s) result in an error? Which one(s)? Are they Type I or Type II errors?

Solution

1) 95% confidence interval for samples of size 10 is (21.67073304,41.06079064)

    95% confidence interval for samples of size 30 is (25.0736473,35.41023)

    95% confidence interval for samples of size 100 is (28.01062,33.99666378)

2) 99% confidence interval for samples of size 10 is (18.60393822,44.12758547)

   99% confidence interval for samples of size 30 is (23.438779,37.0451)

    99% confidence interval for samples of size 100 is (27.06385,34.94343628)

3) all the confidence intervals contains the true population mean

4) samples of size 30 and 100 are significant

5) the 95% confidence interval of sample of size 100 indicates that the population mean lies in the interval (28.01062,33.99666378) with 95% confidence.

6) the 95% confidence interval of sample of size 100 indicates that the population mean lies in the interval (27.06385,34.94343628) with 95% confidence

7) if p-value is less than the significance level then we reject the null hypothesis