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Introductory Statistics Explained (1.11) Exercises Inference for Proportions © 2022, 2023, 2024 Jeremy Balka Chapter 11: Inference for one or two proportions. v1.11 W24 Draft J.B.’s strongly suggested exercises: 2 , 5 , 6 , 8 , 10 , 12 , 16 , 17 , 19 , 20 , 24 , 25 , 26 , 27 NB The section titles and numbers are not yet synced up with the text. 1 Introduction 2 The Sampling Distribution of ˆ p 1. What are the mean and variance of the sampling distribution of ˆ p ? 2. For what values of n and p does the normal approximation to the distribution of ˆ p work best? For what values of n and p is the normal approximation very poor? 3 Confidence Intervals and Hypothesis Tests for the Popula- tion Proportion p 3. When do we create a confidence interval for ˆ p ? 4. A random sample of 200 observations from a population revealed that 62 individuals had a certain characteristic. (a) What is ˆ p ? (b) What is SE (ˆ p ) ? (c) What is a 95% confidence interval for p ? (d) If we wish to test the null hypothesis that p = 0 . 5 , what is SE 0 (ˆ p ) ? (e) What is the value of the appropriate test statistic? 5. In each of the following scenarios, state if it is reasonable to use the normal approximation to calculate a confidence interval for p . (Use the n ˆ p ≥ 15 , n (1 - ˆ p ) ≥ 15 guideline.) (a) ˆ p = 0 . 70 , n = 10 . (b) ˆ p = 0 . 70 , n = 200 . (c) ˆ p = 0 . 99 , n = 200 . 1 Best n is large Das worst nis small p close to 0 or 1 Nevercan only do CI for parameters up to 0.7 7 nil p roll on 3 not both is so donot use normal approx

(d) ˆ p = 0 . 01 , n = 10,000. (e) ˆ p = 0 , n = 200 . 6. A study 1 of births in Liverpool, UK, investigated a possible relationship between parental smoking status during pregnancy and the likelihood of a male birth. In one part of the study, researchers drew a sample of 363 births in which both parents were heavy smokers during the pregnancy. Of the 363 babies born to these couples, 158 were male. Suppose we wish to construct a 95% confidence interval for the population proportion of male births to heavy-smoking parents in Liverpool. (a) What does p represent in this case? (b) What is the point estimate of p ? (c) Is it reasonable to use large sample methods to calculate a confidence interval for p here? (d) What is the standard error of the sample proportion? (e) What is a 95% confidence interval for p ? (f) Give an interpretation of the 95% confidence interval for p . (g) Test the null hypothesis that the population proportion of male births to heavy-smoking parents is 0.50, against a two sided alternative. Give appropriate hypotheses, standard error, value of the test statistic, p -value, and conclusion at ↵ = 0 . 05 . (h) To what population do your conclusions apply? 4 Determining the Minimum Sample Size n 7. Find the minimum sample size required in each of the following situations. (a) We wish to estimate p within 0.03 with 95% confidence, and we have no reasonable estimate of p beforehand. (b) We wish to estimate p within 0.03 with 95% confidence, and from prior information we feel strongly that p is approximately 0.20. (c) We wish to estimate p within 0.01 with 90% confidence, and we have no reasonable estimate of p beforehand. (d) We wish to estimate p within 0.01 with 99% confidence, and we have no reasonable estimate of p beforehand. (e) We wish to estimate p within 0.01 with 99% confidence, and we know for certain that p lies between 0.1 and 0.2. 5 Inference Procedures for the Di ff erence Between Two Pop- ulation Proportions 5.1 The Sampling Distribution of ˆ p 1 - ˆ p 2 5.2 Confidence Intervals and Hypothesis Tests for p 1 - p 2 8. In words, what is the meaning of SE (ˆ p 1 - ˆ p 2 ) ? 9. A random sample of 400 observations from population 1 revealed that 82 individuals had a certain characteristic. A random sample of 400 observations from population 2 revealed that 104 individuals had that characteristic. 1 Koshy et al. (2010). Parental smoking and increased likelihood of female births. Annals of Human Biology , 37(6):789–800. np o so 100000000 g 178 Effort gg male births to smokers in Liverpool in time frame or study tassen s on t's Ho p as Ha p prove normal 2.48111 2 I I I

(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