3. (This problem should be completed by hand, except when finding p-values or critical values. This problem also requires the use of concepts that were tested on Exam 2.) Hutchinson-Gilford progeria syndrome is a rare genetic condition that produces rapid aging in children. As a result, cardiovascular disease is a common cause of death in the teenage years. A clinical study on simple random sample of 18 children diagnosed with progeria is designed to examine the effect of treatment with the drug Lonafarnib on the pulse wave velocity (PWV), measured in meters per second (m/s). PWV is the standard measure of vascular stiffness, an important factor in cardiovascular health. Higher values of PWV serve as a greater concern for future health issues.
- Prior to receiving the treatment, the 18 children diagnosed with progeria had a mean PWV of 12.44 m/s with a standard deviation of 3.638 m/s. Assuming the sample size is large enough for the Central Limit Theorem to apply, find a 95% confidence interval for the true mean PWV of children with progeria. Be sure to write down the formula and include your work.
(b) In healthy children (i.e., those without progeria), the true mean PWV is 6.6 m/s. Suppose we wish to use the sample in (a) to investigate whether children with progeria have a higher mean PWV than children without progeria.
(i) State the null and alternative hypotheses.
(ii) Write the formula for the test statistic used to investigate this claim. Options include: one-sample z-test for population mean, one-sample t-test for population mean, two-sample t-test for comparing means of independent samples (equal variance), two-sample t-test for comparing means of independent samples (unequal variance), or matched pairs t-test.
(iii) Compute the test statistic.
(iv) What is the distribution of the test statistic under the null hypothesis? (This is a fancy way of asking for which distribution you would use to find the p-value.)
(v) Make a sketch and shade the area corresponding to the
p-value.
(vi) Without knowing the
p-value, do you think you will reject the null hypothesis?
Explain your reasoning
.
(vii) Suppose the
p-value is
(c) Suppose the 18 children from part (a) are now given Lonafarnib, an enzyme inhibitor that interferes with the mutated protein progeria responsible the syndrome. After receiving a daily dose for two years, the PWV levels are reassessed. To investigate the effectiveness of the drug, the improvement is found by computing PWVafter
- PWVbefore
for each child in the study.
(i) If the drug is effective, would you expect the differences to be positive or negative?
(ii) If we want to perform inference (say, a hypothesis test) on the effectiveness of the drug, which test statistic would you use? Options include: one-sample z-test for population mean, one-sample t-test for population mean, two-sample t-test for comparing means of independent samples (equal variance), two-sample t-test for comparing means of independent samples (unequal variance), or matched pairs t-test.
(iii) State the null and alternative hypotheses.
(iv) In the sample, the mean difference (PWVafter
- PWVbefore) was -3.917 m/s with a standard deviation of 2.644 m/s. Compute the test statistic.
(v) What is the distribution of the test statistic under the null hypothesis? (Asking for which distribution you use to find the p-value.)
(vi) Make a sketch and shade the area corresponding to the
p-value.
(vii) Without knowing the
p-value, do you think you will reject the null hypothesis?
Explain your reasoning
.
(viii) Suppose the
p-value is