Chapter 9: Inferences on Proportions WB9-01: In a random sample of 1000 voters, 538 favor candidate A. What is a 99% confidence interval for p, the true proportion of all voters who favor A? WB9-02: In the survey example in WB9-01, how large must n be so that the 99% confidence interval for p has the difference, d, between p and p as at most 0.03? WB9-03: In a random sample of 250 high school seniors in a Midwestern state, 168 said they intended to continue their education at an in-state college or university. Find a 95% confidence interval for the corresponding true proportion."-""J--'rw n of drivers who speed on a certain I EA. How large a sampie must we take n than 0.041! n. WEB-04: Suppose we want to estimate the meOI'I-io stretch of road between Los Males and Bakerseid, order to be 909i. condent our estimate Is off Inf "0 "1"! _________,__.......-... _ a} ...a similar survey in 1985 indicated that 65% of people speed; 1:) \"we have no information from a previous survey? 5939-05: A city council member claims that 45% of her constltue ncv is "very concerned'r about drug trafcking. To see if this claim is too low. a sample 01'265 citizens was taken and 135 of them indicated that they were "veryI concerned" about drug trafcking. Test at the 0.05 level, WEB-06: A local newspaper claims that 2595 of its readers regularly clip coupons from the newspaper. To see if this number is reasonable, 185 readers were surveyed and M of them indicated their regulariv clipped coupons. Conduct an appropriate test allowing a 10% Type I error rate. \fWB9-09: Superplasticized concrete is formed by adding chemicals to conventional concrete to make it more fluid so that it can be placed more easily. Suppose that a sample of 50 new construction projects in the Dallas-Fort Worth area yields 15 that are using this type of concrete. A sample of 60 new projects in the Boston area also yields 15 using superplasticized concrete. a) Let po and ps denote the proportion of new concrete projects in Dallas-Fort Worth and Boston, respectively, that are using superplasticized concrete. Find point estimates for PD, PB, and PD - PB. b) Find a 95% confidence interval for PD - PB. c) Would you be surprised to hear someone claim that the proportion of Dallas-Fort Worth projects using this type of concrete is clearly larger than that in the Boston area? Explain, based on the confidence interval in from part b. d) What common sample size must we take from the populations of Dallas-Fort Worth and Boston projects to estimate po - ps to within 0.02 with 90% confidence? Use the point estimates in a) as prior estimates.W59 10- We wish to compare proportions of defectives for two processes; Indepegndem random samples are taken with n; = 1000 x: = 50 and n: = 1250 Kzf: 4:; :V altIs a? 0% condence interval for the true difference in the proportion of do at e , p; p: was-11: With at = 0.05, we wish to determine ffthe proportion of defectives for Process 2 is signicantly less than that for Process 1. Using the data given in Wag-10, test the hypothesis. WBQIZ: A study is to be conducted to estimate the difference in the proportion of defective items produced during two different shifts of 3532mny line workers. What common sample size should be used to estimate this difference to within 0.04 with 90% confidence? \""3943: A company is experimenting with a new method of etching circuits that should decrease the proportion of circuits that must be etched a second time. To be cost effect, the difference in proportions between the old and new methods must exceed o_1_ A random sample was taken of 25 circuits that were etched with the old method and 4 were found to need a second etching. A random sample of 50 circuits etched with the new method was taken and 2 were found to need a second etching. a} Let po and pr. denote the proportion of circuits using the old and new etching methods, respectively. that need etching a second time. Set up the null and alternative hypotheses required to show that the new method is cost effective. b} Find the critical value for o. = 0.05 level of signicance. c] Perform the test. Can He he rejected at the 0.05 level? d} To what type of error are you now subject to? e] What are the practical consequences of making such an error? WEB-14: One measure of quality control and customer satisfaction is repeat business. A supplier of paper used for computer printouts sampled 75 customer accounts last year and found that 40 of these had placed more than one order during the year. A similar survey conducted at the end of the current year revealed 35 out of 50 customers ordered again. Do these data support the contention that there has been an increase in the proportion of repeat business overthe 2-year period? Test at the 0.05 level of signicance. \fdata in 1909. of those com! convicted of fraud, 65 were drinkers and 1M abstained Use Mlnitab output shown below to test the claim that the proportlon of drinkers among co arsanists is greater than the proportion of drinkers convicted of fraud. Does it seem reasonable that drinking might have had an at on the type of rlme? Why? [T no a, Elementarv Statistics, 8*\" Edition, page 468} for Two Preportions WB9-17: When games were sampled from throughout a season, it was found that the home team won 127 of 198 professional basketball games, and the home team won 57 of 99 professional football games (based on data from "Predicting Professional Sports Game Outcomes from Intermediate Game Scores," by Cooper, et al., Chance, Vol. 5, No. 3-4). Using the 0.05 level of significance, test the claim that there is a difference in proportions for home court/field game wins. (Adapted from Triola, Elementary Statistics, 8" Edition, page 468) Test and CI for Two Proportions Method P1: proportion where Sample 1 = Basketball P2: proportion where Sample 2 = Football Difference: P1 - Pz Descriptive Statistics Sample N Event Sample p Basketball 198 127 0.641414 Football 99 57 0.575758 Estimation for Difference 95% Cl for Difference Difference 0.0656566 (-0.052412, 0.183726) CI based on normal approximation Test Null hypothesis Ho: P1 - Pz = 0 Alternative hypothesis H1: p1 - P2 # 0 Method Z-Value P-Value Normal approximation 1.10 0.272 The pooled estimate of the proportion (0.619529) is used for the tests