A researcher is studying the craniofacial morphology of patients diagnosed with

Question

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?

20.57194592 21.83298134 34.42427843 14.53339286 8.072751784 26.98182352 46.03740508 34.21397925 37.40266971 58.79257461 40.04836234 67.80208995 18.81920563 17.62781401 33.59429062 37.60562095 34.07513617 18.16696087 45.36587936 25.88495957 24.28507952 9.386280678 15.56885697 33.99671697 27.75073739 33.37652929 15.3015488 16.22177593 24.31583938 55.53073561 20.40754448 32.26777311 19.70742968 50.0131846 44.2109054 54.43077566 24.39000716 30.31214416 78.14632887 12.72852392 13.69279988 31.16620223 8.758950706 24.95874873 9.667481554 26.17498671 23.79462395 37.74721684 53.48763703 30.77232158 33.78089098 33.45882286 48.02791183 14.07981768 77.69977971 30.74557395 47.22636244 35.54109118 16.91526062 33.11402076 24.45597441 34.52416483 7.80282322 18.70475462 45.33733738 29.55233827 21.26137392 15.69269004 36.30567258 30.03600408 43.46550066 30.931435 17.87110937 23.2108371 23.91518653 35.03367665 28.06112039 50.3767829 7.497446952 57.69004937 24.90610411 71.51276554 20.34658378 21.60707958 47.89853542 22.561601 26.44214977 9.565718705 38.61884899 12.666107 12.68144011 42.86712839 9.77372061 31.82262897 47.99678358 33.37002341 41.65235006 35.09312841 35.63990038 26.17019396

Explanation / Answer

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

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