11. exercises_proportions

(a) What are the values of ˆ p 1 and ˆ p 2 ? (b) What is SE (ˆ p 1 - ˆ p 2 ) ? (c) Calculate a 95% confidence interval for p 1 - p 2 . (d) Test the null hypothesis that the population proportions are equal, against a two-sided alter- native hypothesis. Give appropriate hypotheses, value of the pooled proportion ˆ p , value of the standard error, test statistic and p -value. Is there significant evidence against the null hypothesis at ↵ = 0 . 05 ? 10. Much research has gone into studying how homing pigeons are able to navigate to their home loft from an unfamiliar release point, but the precise mechanisms are still unknown. It is known that homing pigeons can detect the earth’s magnetic field, and that is likely a contributing factor in their ability to navigate. A study 2 investigated this in an experiment involving 77 homing pigeons. The pigeons were randomly divided into a magnetic pulse group (group M) and a control group (group C). The 38 control group pigeons were released at a location 106 km from the home loft, and 22 found their way home. The 39 members of the magnetic pulse group received a strong magnetic pulse (perpendicular to the earth’s magnetic field) before being released from the same location. Twenty-one of the 39 magnetic pulse group pigeons made it back to the home loft. (a) Calculate a 95% confidence interval for the di ff erence between the population proportion of pigeons that navigate home after being subjected to the magnetic pulse, and the population proportion for the control group. (Hint to ease the calculation burden: SE (ˆ p M - ˆ p C ) = 0 . 1131 .) (b) Give an appropriate interpretation of the interval found in 10a . (c) Perform a hypothesis test of the null hypothesis that the two groups have the same likeli- hood of making it back to their home loft. Give appropriate hypotheses, value of the test statistic, p -value, and conclusion. (One could make an argument for the one-sided alterna- tive hypothesis that the magnetic pulse reduces the probability of a pigeon arriving home, but play it safe and use a two-sided alternative hypothesis.) (Hint to ease the calculation burden: SE 0 (ˆ p M - ˆ p C ) = 0 . 1132 .) 6 Chapter Exercises 6.1 Basic Calculations 11. Calculate the p -value in the following situations. (a) H 0 : p = 0 . 3 , H a : p > 0 . 3 , Z = - 1 . 40 . (b) H 0 : p = 0 . 3 , H a : p < 0 . 3 , Z = - 1 . 40 . (c) H 0 : p = 0 . 3 , H a : p 6 = 0 . 3 , Z = - 1 . 40 . (d) H 0 : p = 0 . 6 , H a : p < 0 . 6 , Z = - 1 . 88 . (e) H 0 : p = 0 . 6 , H a : p 6 = 0 . 6 , Z = - 1 . 88 . 6.2 Concepts 12. What is the di ff erence in meaning of the symbols ˆ p and p ? 13. Would it ever make sense to test H 0 : ˆ p = 0 . 25 ? 14. What is the meaning of the term SE (ˆ p ) ? Does the term SE ( p ) have a similar meaning? 2 Holland et al. (2013). A magnetic pulse does not a ff ect homing pigeon navigation: a gps tracking experiment. The Journal of Experimental Biology , 216:2192–2200. Pc 22138 0 s.IE.IE iEiiE i ii ii iii iii iii iii iiiiiii.fi no evidence be small prime iii iii iii p.si proportion p the pñ portion for entire population ly Nowe test hypothesis about parameters not statistics estima

(c) Perform a hypothesis test of the null hypothesis that, in Finnish murders, the proportion of victims that are the domestic partner of the o ff ender is the same for male and female o ff enders, against the alternative that these proportions are di ff erent. Give appropriate hypotheses, value of the test statistic, p -value, and conclusion at the 5% level of significance. (Hint to ease the calculation burden: SE 0 (ˆ p F - ˆ p M ) = 0 . 0684 .) 23. A study 5 at a medical clinic in Iran investigated a possible association between peptic ulcers and a variety of factors. The study involved a sample of 60 peptic ulcer su ff erers from a medical clinic, and a sample of 44 apparently healthy volunteers to serve as the control group. In one part of the study, the authors investigated a possible di ff erence between the groups in their Lewis B (Le b ) blood group antigen expression. (Previous studies had shown a possible relationship between Le b expression and a mechanism for peptic ulcers.) Forty-three of the 60 peptic ulcer su ff erers had Le b expression. Twenty-seven of the 44 control group members had Le b expression. (a) Calculate a 95% confidence interval for the di ff erence in the population proportions between peptic ulcer patients and the control group. (Hint to ease the calculation burden: SE (ˆ p U - ˆ p C ) = 0 . 0937 .) (b) Give an appropriate interpretation of the interval found in 23a . (c) Perform a hypothesis test of the null hypothesis that the proportion of individuals with Le b expression is the same for the two groups, against the alternative hypothesis that the proportions are di ff erent. Give appropriate hypotheses, value of the test statistic, p -value, and conclusion at the 5% level of significance. (Hint to ease the calculation burden: SE 0 (ˆ p U - ˆ p C ) = 0 . 0931 .) 24. Consider again the study first discussed in Question 6 , which investigated a possible relationship between parental smoking status during pregnancy and the likelihood of a male birth. Of 363 births in which both parents were heavy smokers during the pregnancy, 158 of the babies were male. Of the 5045 births in which both parents were non-smokers during the pregnancy, 2685 of the babies were male. (a) Calculate a 95% confidence interval for the di ff erence between the population proportion of male births for heavy-smoking parents, and the population proportion of male births for non-smoking parents. (Hint to ease the calculation burden: SE (ˆ p H - ˆ p N ) = 0 . 0270 .) (b) Give an appropriate interpretation of the interval found in 24a . (c) Perform a hypothesis test of the null hypothesis that the male birth rate is the same for both heavy-smoking and non-smoking parents. Give appropriate hypotheses, value of the test statis- tic, p -value, and conclusion at the 5% level of significance. (Hint to ease the calculation burden: SE 0 (ˆ p H - ˆ p N ) = 0 . 0271 .) (d) To what population do your conclusions apply? Do the results of this study imply strong evidence that parents smoking cause a decrease in the proportion of male births? 25. Does a pleasant ambient fragrance increase the success rate of a courtship request? A study 6 in- vestigated this question by having a good-looking 20 year-old man approach 18-25 year-old women walking alone in a mall. On 200 occasions, the man approached as the woman was walking in an area of “pleasant ambient odours (e.g., pastries)”, and on 200 occasions he approached in an area of no odour. In every case, the man approached the woman, made a brief scripted statement including a compliment, and then asked for the woman’s phone number. Of the 200 cases in areas of pleasant ambient odour, 46 women gave their phone number. Of the 200 cases in areas of no ambient odour, 27 women gave their phone number. 5 Mahdavi et al. (2013). Is there any relationship between leb antigen expression and Helicobacter pylori infection? Blood Cells, Molecules and Diseases , 51:174–176. 6 Guéguen et al. (2012). The sweet smell of. courtship: E ff ects of pleasant ambient fragrance on womenÕs receptivity to a manÕs courtship request. Journal of Environmental Psychology , 32:123–125 I 461200 0.23 PI 27 200 0.0135

(c) Perform a hypothesis test of the null hypothesis that the approval ratings for men and women are the same, against the alternative that they are di ff erent. Give appropriate hypotheses, value of the test statistic, p -value, and a conclusion at the 5% level of significance. (Hint to ease the calculation burden: SE 0 (ˆ p F - ˆ p M ) = 0 . 070707 .) 33. A company is testing a new heat treatment intended to eradicate bedbugs. As part of the investi- gation process, they wish to assess what e ff ect di ff erent temperatures have on killing bedbugs. They subject 200 adult bedbugs to a temperature of 45 degrees Celsius for 30 minutes, and find that 176 died. They subject 200 di ff erent adult bedbugs to a temperature of 47 degrees Celsius for 30 minutes, and find that 184 died. (a) What is a 95% confidence interval for the di ff erence in population proportions? Also give an appropriate interpretation of the interval. (Hint to ease the calculation burden: SE (ˆ p 1 - ˆ p 2 ) = 0 . 02993 .) (b) Suppose we wish to test the null hypothesis that the population proportions are equal, against the alternative that the higher temperature kills a higher proportion of bedbugs. Give appropri- ate hypotheses in words and symbols, standard error, value of the test statistic, and conclusion at a ↵ = 0 . 01 significance level. (Hint to ease the calculation burden: SE 0 (ˆ p 1 - ˆ p 2 ) = 0 . 030 .) References Chase et al. (2007). Safety restraint injuries in fatal motor vehicle collisions. Forensic Science, Medicine, and Pathology , 3:258–263. Guéguen et al. (2012). The sweet smell of. courtship: E ff ects of pleasant ambient fragrance on womenÕs receptivity to a manÕs courtship request. Journal of Environmental Psychology , 32:123–125. Häkkänen et al. (2009). Gender di ff erences in Finnish homicide o ff ence characteristics. Forensic Science International , 186:75–80. Holland et al. (2013). A magnetic pulse does not a ff ect homing pigeon navigation: a gps tracking experi- ment. The Journal of Experimental Biology , 216:2192–2200. Koshy et al. (2010). Parental smoking and increased likelihood of female births. Annals of Human Biology , 37(6):789–800. Mahdavi et al. (2013). Is there any relationship between leb antigen expression and Helicobacter pylori infection? Blood Cells, Molecules and Diseases , 51:174–176.