. Population proportion (parameter) = p.
. Population proportion (parameter) = p.
9.1 Incidence of improvement
(A) Calculate the
proportion in the sample.
= 20 / 75 = 0.2667 (about 27%)
(B) Calculate a 95% confidence interval for the incidence in the population. n~ = 79, X~ = 22, p~ = 0.2785, SEp~ = sqrt[(0.2785)(0.7215)/(79)] = 0.0504; 95% CI for p = 0.2785 ± (1.96)(0.0504) = 0.2785 ± 0.0988 = (0.1797 to 0.3773) or (18% to 37%).
(C) Explain the meaning of the confidence interval to a person who has little statistical training.
In explaining this confidence interval to a lay person, you should be clear that the sample proportion is only a rough estimate of the
true proportion in the population. The confidence interval gives us a better idea of this true value. In this case, we can say with 95% confidence that the true proportion is between 18% to 37%.
(D) How large a sample is needed to reduce the margin of error of the 95% confidence
interval to plus or minus 0.05? In using the proportion from the initial study
as p*, we calculate n = (1.962)(0.267)(0.733) / (0.052)
= 300.7. Round this up to the next integer, to use n = 301.
9.3 BRCA1 mutation:
= 0.160. 95% CI for p = 0.112 to 0.222 (by Wilson's method, which
should be similar to the plus-four results)
9.5 Risk factor X:
= 16 / 120 = 0.1333 ( about 13%); 95% confidence interval for p = 0.084 to 0.206 (via
Wilson's method, which should be similar to the plus-four method)
9.7 Insulation workers Test whether the observed number of cases is significantly greater than expected. Show all hypothesis testing steps.
Check the npq rule: np0q0 = (556)(0.0259)(1 - 0.0259) = 14.0. Then...
(Hypotheses) H0: p = 0.0259 vs. H1: p not = 0.0259
(Test statistic) SE = sqrt[(0.02590)(1 - 0.02590) / (556)] = 0.00674;
(P-value) P = 0.0020
(Significance statement). The evidence against H0 is highly significant.
9.9 Leukemia gender preference --The z test can be use since: np0q0 = (262)(0.5)(0.5) = 65.5.
Check the npq rule: np0q0 = (262)(0.5)(1 - 0.5) = 65.5
(Hypotheses) H0: p = 0.5 vs. H1: p0.5
(Test statistic): SE = sqrt[(0.5)(0.5)/(262)] = 0.03089; zstat = (0.5725 - 0.5) / 0.03089 = 2.34
(P-value) P = 0.0192 indicating good evidence against H0
(Significant statement): Data provide significant evidence against H0.
9.11 Sample size requirements
(A) n = (1.962)(0.5)(0.5) / 0.12 = 96.04. Round this up to n = 97 to ensure adequate precision.
(B) n = (1.962)(0.5)(0.5) / 0.052 = 384.16. Round up to n = 385.
(C) n = (1.962)(0.5)(0.5) / 0.0252 = 1536.64, so use n = 1537.
9.13 Alternative medicine. 95% confidence interval for parameter p = 0.415 to 0.465 (42% to 47%) by the Wilson score method.
9.15 Cerebral tumors -- same side as cell phone use? p-hat = 26 / 41 = 0.6341, q-hat = 1 - 0.6341 = 0.3659, SE (assuming p = 0.5) is equal to sqrt(.5*.5/41) = 0.078087. zstat = (0.6351 - .5)/ 0.078087 = 1.718; one-sided P = Pr(Z >= 1.718) = 0.043; two-sided P = 2 * 0.43 = 0.086.
Notes for advanced users:
9.17 Drove when drinking alcohol.
(A) The extent to non-response biased the survey depends on if those who refuse to participate differed from those who participated in the study.
(B) No. A repeatable response can be repeatedly wrong. Some things to consider regarding accuracy: Did the respondents understand the question? Were they fearful of telling the truth about their behaviors? Were they trying to please the people administering the survey? (And so on.)