Comparing 2 Proportions

Worksheet 1

Exercises

  1. High levels of cholesterol in the blood are associated with higher risk of heart attacks. Will using a drug to lower blood cholesterol reduce heart attacks? The Helsinki Heart Study looked at this question. Middle-aged men were assigned at random to one of two treatments: 2051 men took the drug gemfibrozil to reduce their blood cholesterol levels, and a control group of 2030 men took a placebo. During the next five years, 56 men in the gemfibrozil group and 84 men in the placebo group had heart attacks.
    1. Produce an appropriate display of the results.
    2. Is there a significant difference between the heart attack rate of those who took gemfibrozil and the rate of those who received the placebo? Test the appropriate hypotheses at the 1% significance level. Interpret the P-value.
  2. A study of the survival of small businesses chose a random sample from the telephone directory’s Yellow Pages listings. Interviews were completed with 148 businesses; 106 were headed by men and 42 by women. Three years later, 22 of those businesses had failed: 15 headed by men, 7 by women.
    1. Produce an appropriate display of the study results.
    2. Is there a significant difference between the rate at which businesses headed by men and those headed by women fail? Formulate and test the appropriate hypotheses at the 5% level. Report the value of the test statistic as well as the P-value and your decision.
    3. Report also a 95% CI. Write one complete sentence expressing the result.
  3. A study of the survival of small businesses chose a random sample from the telephone directory’s Yellow Pages listings. Interviews were completed with 4440 businesses; 3180 were headed by men and 1260 by women. Three years later, 660 of those businesses had failed: 450 headed by men, 210 by women.
    1. Produce an appropriate display of the study results.
    2. Is there a significant difference between the rate at which businesses headed by men and those headed by women fail? Formulate and test the appropriate hypotheses at the 5% level. Report the value of the test statistic as well as the P-value and your decision.
    3. Report also a 95% CI. Write one complete sentence expressing the result.
  4. Compare solutions to the previous two exercises. Note how similar (in one respect) the two exercises are, yet how different some of the results are.
  5. Two drugs (call them A and B) are available to Emergency Medical Technicians (EMTs) treating cardiac arrest. The drugs are generally administered in an ambulance. Both drugs are quite effective—people usually survive (not only because of the drugs, but at least in part so) when either of these drugs is used soon after an attack. A clinical study compares the effectiveness of these drugs. As EMTs drive to a patient, a computer randomly determines which drug will be administered to that patient. At the conclusion of the study, 4931 women have been treated with drug A of whom 4854 recovered; 4876 were treated with drug B of whom 4764 survived.
    1. Is there evidence of a difference between the survival rates of patients receiving the two drugs? Formulate the appropriate hypotheses and test at the 1% level. Report the value of the test statistic as well as the P-value and your decision.
    2. Obtain a 95% CI for the difference between the survival rate for drug A and that for drug B. Write one complete sentence expressing the result.
    3. If you’ve answered part (a) correctly, you’ve learned that the difference is statistically significant at the 1% level (P-value = 0.008); part (b) tells us (with 95% confidence) that the survival rate for drug A is between 0.2% and 1.3% higher than it is for drug B. Now: You are one of the hospital managers. You treat about 50 such patients per year. Over a 10 year period, how many would you expect to die using only drug A? Using only drug B? (After 10 years, there will probably be a better drug than both of these.) The pressure is on the management to “keep costs down.” Drug A costs $1000 per dose to administer; drug B costs $100. How much money is saved over this 10-year period if drug B is used instead of drug A? Which should be used? What if drug B costs $2000 per dose? $3000? At what price point would you have the hospital use drug B? Given the choice, and assuming you were the patient, which drug do you want to be treated with?

Solutions

  1.  

    1. Treatment       Percentage having heart
      attacks (number treated)
      Gemfibrozil  2.73% (2051)
      Placebo 4.14% (2030)
    2. H0: p1 - p2 = 0   HA: p1 - p2 < 0 where p1 and p2 are the (population) proportions of people treated with gemfibrozil and placebo, respectively, who have a heart attack within a five year period years. The test statistic is -2.47; P-value 0.007. The probability of obtaining a difference as large as is the observed difference is 0.007 (1 in 143). There is rather strong evidence for the alternative hypothesis which states that gemfibrozil improves one's chances of a heart attack. At the 1% level we reject H0.
  2.  

    1. Sex of Business Owner      Percent of businesses
      failing (number surveyed)
      Man 14.2% (106)
      Woman 16.7% (42)
    2. H0: p1 - p2 = 0   HA: p1 - p2 ¹ 0 where p1 and p2 are the (population) failure rates for male and female small businesses, respectively. The test statistic is -0.39; P-value 0.698. The observed difference is the sort of difference that would be observed 7 in 10 times just due to chance alone. Fail to reject H0 at the 5% level.
    3. -2.5% ± 13.1% or (-15.6%, 10.6%). With 95% confidence we estimate that the failure rate for men is between 15.6% lower and 10.6% higher than it is for women.
  3.  
    1. The %s are the same as in (2); only the sample sizes change -- to 3180 and 1260.
    2. The hypotheses are the same as in (2). Here the test statistic is -2.12 and P-value is 0.034. Reject H0 at the 5% level. There is a difference -- and it's women who's businesses are (slightly; see (c) below) more likely to fail.
    3. -2.5% ± 2.4% or (-4.9%, -0.1%). With 95% confidence we estimate that the failure rate for men is between 0.01% and 4.9% lower than it is for women.
  4.  
  5.  
    1. H0: p1 - p2 = 0   HA: p1 - p2 ¹ 0 where p1 and p2 are the (population) recovery rates for the two drugs. The test statistic is 2.65; P-value 0.008. At the 1% level H0: is rejected; there is a significant difference -- drug A is more effective than is drug B.
    2. 0.74% ± 0.55% or (0.19%, 1.28%). With 95% confidence we estimate that between .19% and 1.28% more patients will survive when given drug A rather than drug B.
    3. Using drug A we'd expect about 8 deaths; about 11 with drug B. On the other hand, drug A will cost one-half million dollars, while drug B will only run $50,000; using drug B would save about $450,000 over ten years. (For one hospital this might not matter, but over the entire country it would add up to a staggering amount. And, that amount only gets larger when the slightly better drug costs even more.)