1. True or False: The null hypothesis can be summarized as nothing is going on besides chance
variation."
2. In the aspirin study with strokes as the outcome, the 95% con dence interval was (.93,1.59)
and the p-value was .16. True or false: There is a 16% probability that the null hypothesis is
true.
4.
(a) If the sample size goes up, what will happen to the width of the confidence interval?
(b) If ^p moves closer to 0 or 1, will the confidence interval get wider or narrower?
(c) What is wider, a 95% confidence interval or a 90% con dence interval?
5. In a 2006 study published in The New England Journal of Medicine, 78 pairs
of patients with Parkinson's disease were randomly assigned to receive treatment (which
consisted of deep-brain stimulation of a region of the brain affected by the disease) or control
(which consisted of taking a prescription drug). The researchers found that in 50 of 78 pairs,
the patients who received deep-brain stimulation had improved more than their partner in
the control group. We are interested in conducting a hypothesis test of these findings.
(a) What is the null hypothesis?
(b) Let x denote the number of pairs in which the treatment patient did better than the
control patient. If the null distribution were true, what distribution would x follow? If
this distribution has parameters, what values do they have?
(c) Calculate a p-value for this experiment using a sign test.
(d) Calculate a p-value for this experiment using a z-test.
(e) Do these hypothesis tests provide strong evidence against the null hypothesis?
(f) Construct (by hand) an approximate 95% confidence interval for the percentage of pa-
tients with Parkinson's disease in the population who will do better with deep-brain
stimulation than with the control drug.
(g) Construct (using a computer) an exact 95% confidence interval for the percentage of
patients with Parkinson's disease in the population who will do better with deep-brain
stimulation than with the control drug.
(h) In the paper, the authors claimed that deep-brain stimulation was more effective than
medical management." Based on your analyses, would you agree or disagree with their
findings?
6. A crossover study published in the American Journal of Clinical Nutrition investigated
whether oat bran cereal helps to lower serum cholesterol levels. Fourteen individuals were
placed on a diet that included either oat bran or corn flakes; after two weeks, their LDL
cholesterol levels were measured. Each individual was then switched to the other diet; after
two weeks, the LDL levels were recorded again. The data from the study are on the course
website.
(a) If the investigators had decided to perform a study that consisted of randomizing two
separate groups of men to the two diets, would the power of the study been (i) lower (ii)
higher (iii) the same?
(b) Perform a sign test of the null hypothesis that eating oat bran cereal has no impact on
LDL cholesterol levels.
(c) Perform a z-test of the same hypothesis.
(d) Construct an exact 95% con dence interval for the percentage of individuals with high
cholesterol who would experience a reduction in cholesterol on an oat bran diet vs. a
corn
ake diet.
(e) Construct an approximate 95% con dence interval for the percentage of individuals with
high cholesterol who would experience a reduction in cholesterol on an oat bran diet vs.a corn
flake diet.
8. Recall that when John Kerrich tossed his coin 10,000 times, he got 5,067 heads.
Was Mr. Kerrich tossing a fair coin? Support your conclusion with a hypothesis test.
variation."
2. In the aspirin study with strokes as the outcome, the 95% con dence interval was (.93,1.59)
and the p-value was .16. True or false: There is a 16% probability that the null hypothesis is
true.
4.
(a) If the sample size goes up, what will happen to the width of the confidence interval?
(b) If ^p moves closer to 0 or 1, will the confidence interval get wider or narrower?
(c) What is wider, a 95% confidence interval or a 90% con dence interval?
5. In a 2006 study published in The New England Journal of Medicine, 78 pairs
of patients with Parkinson's disease were randomly assigned to receive treatment (which
consisted of deep-brain stimulation of a region of the brain affected by the disease) or control
(which consisted of taking a prescription drug). The researchers found that in 50 of 78 pairs,
the patients who received deep-brain stimulation had improved more than their partner in
the control group. We are interested in conducting a hypothesis test of these findings.
(a) What is the null hypothesis?
(b) Let x denote the number of pairs in which the treatment patient did better than the
control patient. If the null distribution were true, what distribution would x follow? If
this distribution has parameters, what values do they have?
(c) Calculate a p-value for this experiment using a sign test.
(d) Calculate a p-value for this experiment using a z-test.
(e) Do these hypothesis tests provide strong evidence against the null hypothesis?
(f) Construct (by hand) an approximate 95% confidence interval for the percentage of pa-
tients with Parkinson's disease in the population who will do better with deep-brain
stimulation than with the control drug.
(g) Construct (using a computer) an exact 95% confidence interval for the percentage of
patients with Parkinson's disease in the population who will do better with deep-brain
stimulation than with the control drug.
(h) In the paper, the authors claimed that deep-brain stimulation was more effective than
medical management." Based on your analyses, would you agree or disagree with their
findings?
6. A crossover study published in the American Journal of Clinical Nutrition investigated
whether oat bran cereal helps to lower serum cholesterol levels. Fourteen individuals were
placed on a diet that included either oat bran or corn flakes; after two weeks, their LDL
cholesterol levels were measured. Each individual was then switched to the other diet; after
two weeks, the LDL levels were recorded again. The data from the study are on the course
website.
(a) If the investigators had decided to perform a study that consisted of randomizing two
separate groups of men to the two diets, would the power of the study been (i) lower (ii)
higher (iii) the same?
(b) Perform a sign test of the null hypothesis that eating oat bran cereal has no impact on
LDL cholesterol levels.
(c) Perform a z-test of the same hypothesis.
(d) Construct an exact 95% con dence interval for the percentage of individuals with high
cholesterol who would experience a reduction in cholesterol on an oat bran diet vs. a
corn
ake diet.
(e) Construct an approximate 95% con dence interval for the percentage of individuals with
high cholesterol who would experience a reduction in cholesterol on an oat bran diet vs.a corn
flake diet.
8. Recall that when John Kerrich tossed his coin 10,000 times, he got 5,067 heads.
Was Mr. Kerrich tossing a fair coin? Support your conclusion with a hypothesis test.