# Various Hypothesis Testing Problems

Diabetes

The insulin pump is a device that delivers insulin to a diabetic patient at regular intervals. It presumably regulates insulin better than standard injections. However, data to establish this point are few, especially in children. The following study was set up to assess the effect of use of the insulin pump on HgbA1c, which is a long-term marker of compliance with insulin protocols. In general, a normal range for HgbA1c is < 7%. Data were collected on 256 diabetic patients for 1 year before and after using the insulin pump. A subset of the data for 10 diabetic patients is given in Table 8.40 (see attachment).

8.163 What test can be used to compare the mean HgbA1c 1 year before vs. mean HgbA1c 1 year after use of the insulin pump?

8.164 Perform the test in Problem 8.163, and report a two-tailed p- value.

8.165 Provide a 95% CI for the mean difference in HgbA1c before minus the mean HgbA1c after use of the insulin pump.

Cardiology

Example31: Cardiovascular Disease The Physicians' Health Study was a randomized clinical trial, one goal of which was to assess the effect of aspirin in preventing myocardial infarction (MI). Participants were 22,000 male physicians ages 40- 84 and free of cardiovascular disease in 1982. The physicians were randomized to either active aspirin (one white pill containing 325 mg of aspirin taken every other day) or aspirin placebo (one white placebo pill taken every other day). As the study progressed, it was estimated from self- report that 10% of the participants in the aspirin group were not complying (that is, were not taking their study [aspirin] capsules). Thus the dropout rate was 10%. Also, it was estimated from self- report that 5% of the participants in the placebo group were taking aspirin regularly on their own outside the study protocol. Thus the drop- in rate was 5%. The issue is: How does this lack of compliance affect the sample size and power estimates for the study?

(No need to answer anything here. Ex 31 is provided because Ex 32 refers to it)

Example 32: Refer to Example 10.31. Suppose we assume that the incidence of MI is .005 per year among participants who actually take placebo and that aspirin prevents 20% of MIs (i. e., relative risk = p 1 / p 2 = 0.8). We also assume that the duration of the study is 5 years and that the dropout rate in the aspirin group = 10% and the drop- in rate in the placebo group = 5%. How many participants need to be enrolled in each group to achieve 80% power using a two- sided test with significance level = .05?

(No need to answer anything here. Ex 32 is provided because problem 10.1 refers to it)

Consider the Physicians' Health Study data presented in Example 10.32.

10.1 How many participants need to be enrolled in each group to have a 90% chance of detecting a significant difference using a two- sided test with a = .05 if compliance is perfect?

Pulmonary Disease

One important aspect of medical diagnosis is its reproducibility. Suppose that two different doctors examine 100 patients for dyspnea in a respiratory disease clinic and that doctor A diagnosed 15 patients as having dyspnea, doctor B diagnosed 10 patients as having dyspnea, and both doctor A and doctor B diagnosed 7 patients as having dyspnea.

10.16 Compute the Kappa statistic and its standard error regarding reproducibility of the diagnosis of dyspnea in this clinic.

Hepatic Disease

Refer to Data Set HORMONE. DAT, provided as an attachment.

10.37 What test procedure can be used to compare the percentage of hens whose pancreatic secretions increased (post- pre) among the five treatment regimens?

10.38 Implement the test procedure in Problem 10.37, and report a p-value.

Cancer

A topic of current interest is whether abortion is a risk factor for breast cancer. One issue is whether women who have had abortions are comparable to women who have not had abortions in terms of other breast cancer risk factors. One of the best- known breast cancer risk factors is parity (i. e., number of children), with parous women with many children having about a 30% lower risk of breast cancer than nulliparous women (i. e., women with no children). Hence it is important to assess whether the parity distribution of women with and without previous abortions is comparable. The data in Table 10.32 (see attachment) were obtained from the Nurses' Health Study on this issue.

10.61 What test can be performed to compare the parity distribution of women with and without induced abortions?

10.62 Implement the test in Problem 10.61, and report a two- tailed p- value.

Ophthalmology

A 5-year study among 601 participants with retinitis pigmentosa assessed the effects of high- dose vitamin A (15,000 IU per day) and vitamin E (400 IU per day) on the course of their disease. One issue is to what extent supplementation with vitamin A affected their serum- retinol levels. The serum- retinol data in Table 10.33 (see attachment) were obtained over 3 years of follow- up among 73 males taking 15,000 IU/ day of vitamin A (vitamin A group) and among 57 males taking 75 IU/ day of vitamin A (the trace group; this is a negligible amount compared with usual dietary intake of 3000 IU/ day).

10.64 What test can be used to assess whether mean serum retinol has increased over 3 years among subjects in the vitamin A group?

One interesting aspect of the study described in Problem 10.64 is to assess changes in other parameters as a result of supplementation with vitamin A. One quantity of interest is the level of serum triglycerides. Researchers found that among 133 participants in the vitamin A group (males and females combined) who were in the normal range at baseline (< 2.13 µmol/ L), 15 were above the upper limit of normal at each of their last 2 consecutive study visits. Similarly, among 138 participants in the trace group who were in the normal range at baseline (< 2.13 µmol/ L), 2 were above the upper limit of normal at each of their last two consecutive study visits.

10.68 What test can be performed to compare the percent-age of participants who developed abnormal triglyceride levels between the vitamin A group and the trace group?

10.69 Implement the test in Problem 10.68, and report a two- tailed p- value.

Obstetrics

The standard screening test for Down's syndrome is based on a combination of maternal age and the level of serum alpha- fetoprotein. Using this test, 80% of Down's syndrome cases can be identified, while 5% of normals are detected as positive.

10.121 What is the sensitivity and specificity of the test? Suppose that 1 out of 500 infants are born with Down's syndrome.

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#### Solution Summary

This solution is comprised of a detailed explanation on appropriate statistic method used for the problems. Full description is given for the method along with the null and alternative hypotheses is provided in the solution. All formulas are given and all calculations are shown in the solution.

Various First Year Statistic Problems

(See attached file for full problem description)

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For Exercise 1, assume that a sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level.

1. N = 1200, x = 400, 99% confidence

For exercise 2, use the sample data and confidence level to construct the confidence interval estimate of the population proportion p.

2. N = 1200, x = 200, 99% confidence

For exercises 3a and b, use the given data to find the minimum sample size required to estimate a population proportion or percentage. Note: ^ should go "on top" of the p and q - I didn't know how to do it.

3. a. Margin of error: 0.038; confidence level: 95%; p^ and q^ unknown

b. Margin of error: three percentage points; confidence level: 90%; from a prior study, p^ is estimated by the decimal equivalent of 8%

For exercise 4, use the given confidence level and sample data to find (a) the margin of error E and (b) a confidence interval for estimating the population mean μ

4. Starting salaries of college graduates who have taken a statistics course: 95% confidence; n = 28, = $45,678, the population is normally distributed, and σ is known to be $9900

For exercise 5, use the given margin of error; confidence level, and population standard deviation σ to find the minimum sample size required to estimate an unknown population mean μ.

5. Margin of error: 3 lb, confidence level: 99%, σ = 15 lb

For exercise 6a and b, use this 95% confidence interval: (262.09, 374.11). The confidence interval results from using a sample of 80 measured cholesterol levels of randomly selected adults.

6. a. Express the confidence interval in the format of - E < μ < + E.

b. Write a statement that interprets the 95% confidence interval.

7. In order to help identify baby growth patterns that are unusual, we need to construct a confidence interval estimate of the mean head circumference of all babies that are two months old. A random sample of 100 babies is obtained, and the mean head circumference is found to be 40.6 cm. Assuming that the population standard deviation is known to be 1.6 cm, find a 99% confidence interval estimate of the mean head circumference of all two-month-old babies. What aspect of this problem is not realistic?

For problem 8a and b, does one of the following, as appropriate: (a) find the critical value Z /2, (b) find the critical value of t /2, (c) state that neither the normal nor the t distribution applies.

8. a. 90%; n = 9, σ = 4.2; population appears to be very skewed

b. 98%; n = 37; σ is unknown; population appears to be normally distributed

For exercise 9, construct the confidence interval.

9. A study was conducted to estimate hospital costs for accident victims who wore seat belts. Twenty randomly selected cases have a distribution that appears to be bell-shaped with a mean of $9004 and a standard deviation of $5629

a. Construct the 99% confidence interval for the mean of all such costs.

b. If you are a manager for an insurance company that provides lower rates for drivers who wear seat belts, and you want a conservative estimate for a worst case scenario, what amount should you use as the possible hospital cost for an accident victim who wears seat belts?

For problem 10a and b, find the critical z values. In each case, assume that the normal distribution applies.

10. a. = 0.10; H1 is p>0.18

b. = 0.005; H1 is p≠0.20

11. The claim is that the proportion of peas with yellow pods is equal to 0.25 (25%), and the sample statistics include n = 580 peas with 26.2% of them having yellow pods.

For problem 12a and b, state the final conclusion in simple nontechnical terms. Be sure to address the original claim.

12. a. Original claim: The proportion of college graduates who smoke is less than 0.27.

Initial conclusion: Reject the null hypothesis

b. Original claim: The proportion of M&Ms that are blue is equal to 0.10

Initial conclusion: Reject the null hypothesis

For exercises 13a, b, c, d, and e, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that address the original claim. Use the P-value method.

13. a. In a recent year, of the 109,857 arrests for Federal offenses, 29.1% were drug offenses. Use a 0.01 significance level to test the claim that that the drug offense rate is equal to 30%. How can the result be explained, given that 29.1% appears to be so close to 30%?

b. In 1990, 5.8% of job applicants who were tested for drugs failed the test. At the 0.01 significance level, test the claim that the failure rate is now lower if a simple random sample of 1520 current job applicants results in 58 failures. Does the result suggest that fewer job applicants now use drugs?

c. In one study of smokers who tried to quit smoking with nicotine patch therapy, 39 were smoking one year after the treatment, and 32 were not smoking one year after the treatment. Use a 0.10 significance level test to claim that among smokers who try to quit with nicotine patch therapy, the majority are smoking a year after the treatment. Do these results suggest that the nicotine patch therapy is ineffective?

d. The health of the bear population in Yellowstone National Park is monitored by periodic measurements taken from anesthetized bears. A sample of 54 bears has a mean weight of 182.9 lb. Assuming that σ is known to be 121.8 lb, use a 0.10 significance level to test the claim that the population mean of all such bear weights is less than 200 lb.

e. . A random sample of 100 babies is obtained, and the mean head circumference is found to be 40.6 cm. Assuming that the population standard deviation is known to be 1.6 cm, use a 0.05 significance level to test the claim that the mean head circumference of all two-month-old babies is equal to 40.0 cm.

For exercise 14a and b, find the test statistic, P-value, critical value(s), and state the final conclusion.

14. a. Claim: The mean starting salary for college graduates who have taken a statistics course is equal to $46,000. Sample data: n = 65, = $45,678. Assume that σ = $9900 and the significance level is = 0.05

b. Claim: The mean starting salary for college graduates who have taken a statistics course is equal to $46,000. Sample data: n = 27, = $45,678, s = $9900. The significance level is = 0.05

For exercise 15a and b, determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither.

15. a. claim: μ = 75. Sample data: n = 25, = 102, s = 15.3. The sample data appears to come from a population with a distribution that is very far from normal, and σ is unknown.

b. Claim: μ = 2.80. Sample data: n = 150, = 2.88, s = 0.24. The sample data appears to come from a population with a distribution that is not normal, and σ is unknown.

For exercise 16, assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use either the traditional value or P-value method for testing hypotheses.

16. Sugar content for a sample of different cereals is summarized with these statistics: n = 16, = 0.295g, s = 0.168g. Use a 0.05 significance level to test the claim of a cereal lobbyist that the mean for all cereals is less than 0.3g.

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(See attached file for full problem description)

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