3. Consider the distribution of serum cholesterol levels for all 20- to 74-year-

Question

3. Consider the distribution of serum cholesterol levels for all 20- to 74-year-old males living in the United States. The mean of this population is 211 mg/dL, and the standard deviation is 46.0 mg/dL. In a study of a subpopulation of such males who smoke and are hypertensive, it is assumed (not unreasonably) that the distribution of serum cholesterol levels is normally distributed, with unknown mean , but with the same standard deviation as the original population. (a) Formulate the null hypothesis and complementary alternative hypothesis, for testing whether the unknown mean serum cholesterol level of the subpopulation of hypertensive male smokers is equal to the known mean serum cholesterol level of 211 mg/dL of the general population of 20-to 74-year-old males (b) In the study, a random sample of size n-12 hypertensive smokers was selected, and found to have a sample mean cholesterol level of 217 mg/dL. Construct a 95% confidence interval for the true mean cholesterol level of this subpopulation. (c) Calculate the p-value of this sample, at the -.05 significance level (d) Based on your answers in parts (b) and (c), is the null hypothesis rejected in favor of the alternative hypothesis, at the = .05 significance level? Interpret your conclusion: What exactly has been demonstrated, based on the empirical evidence? (e) Determine the 95% acceptance region and complementary rejection region for the null hypothesis. Is this consistent with your findings in part (d)? Why?

Explanation / Answer

Solutiona:

Null hypothesis:

Ho: =211

Alternative Hypothesis:

H1: 211

alpha=0.05

Two tail t test

Solutionb:

n=12

sample mean=217

population std dev=46

95% confidence interval for the true mean choloesterol level of this subpopulation is

sample mean-zcrit(population standard deviation/sqrt(sample size),sample mean+zcrit(population standard deviation/sqrt(sample size)

217-1.96(46/sqrt(12),217+1.96(46/sqrt(12)

=190.97,243.03

lower limit=190.97

upper limit=243.03

We are 95% confident thta the true population mean choloesterol level of this sub population lies in between

190.97 and 243.03

In the confidence interval we have truemean of 211

190.97<211<243.03

Accept nUll hypothesis

Solutionc:

Hypothesi test for single mean --two tail

H0:mean=211

H1:mean not =211

alpha=0.05

t=sample mean-pop mean/sd/sqrt(n)

=211-217/46/sqrt(12)

=-0.452

Df=n-1=12-1=11

the p value is 0.660052

P>0.05

Decison:

Fail to Reject Null hypothesis

Accept Null hypothesis

Solutiond:

Based on answers i parts (b) and (c) the null hypothesis is not rejected in favour of the alternative Hypothesis.

We conclude statitsically at 5% significance level that the unknowm mean serum choloesterol level mu of the sub population of hypertensive male smokers is equal to the knowm mean serum choloesterol level of 211 .

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