Males and females were asked about what they would do if they received a $100 bi

Question

Males and females were asked about what they would do if they received a $100 bill by mail, addressed to their neighbor, but wrongly delivered to them. Would they return it to their neighbor? 69 males and 131 females are surveyed. Use this data set to answer the following questions.

a) By the survey data, what are the proportions of males and females who would return the bill to their neighbors?

b) Construct 90% confidence intervals for the true proportions of males and females who would return the bill respectively.

c) Construct 95% confidence intervals for the true proportions of males and females. Compare the 95% CI with the 90% CI you obtained from b) and discuss how confidence interval affects the width of CI.

***PLEASE ANSWER IN R STUDIO CODE***

male female Yes No Yes Yes Yes Yes Yes No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes No Yes No No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes No Yes Yes Yes No Yes Yes Yes Yes Yes No Yes Yes Yes No Yes No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes No Yes No Yes Yes Yes No Yes Yes Yes No Yes Yes Yes No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes No No

Explanation / Answer

a. 52 males returns the bill to their neighbours

so p^male=52/69=0.75

119 females returned the bill to their neighbours

so 119/131=0.91

b. For 90% z value is 1.645

So E=z*SE, SE=sqrt(pq/n)=0.052 for male

Hence E=1.645*0.052=0.086

So CI=p^+/-E=0.75+/-0.086=(0.664,0.836)

Now 90% of female z value is 1.645

so E=1.645*E, SE=sqrt(pq/n)=0.025, for female

E=0.04

CI=p^+/-E=0.91+/-0.04=(0.87,0.95)

c. Similarly for 95% CI x value for male is 1.96

E=1.96*SE=1.96*0.052=0.10

CI=0.75+/-0.10=(0.65,0.85)

Now for 95% Ci z value is 1.96

so E=1.96*SE, SE=sqrt(pq/n)=0.025

E=0.049

CI=p^+/-E=0.91+/-0.049=(0.86,0.95)

Hence we can see as we increase confidence level width increases.

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