Question

Suppose that Jeff has a US quarter and a US nickel. Jeff believes that his US quarter is more likely to land heads than his US nickel. Suppose Jeff flips his quarter 100 times and gets 51 heads, and Jeff independently flips the nickel 100 times and gets 49 heads.

(a) Construct a 99% confidence interval for the difference in the proportion of times Jeff’s quarter lands heads and the proportion of times Jeff’s nickel lands heads.

(b) If Jeff’s quarter were to truly land heads more often than his nickel, would Jeff hope that the confidence interval in (a) consists of only negative or only positive real numbers? Briefly explain your answer.

(c) What could Jeff do to get a narrower confidence interval than the confidence interval in (a)?

Answer 1

(a)

Therefore, based on the data provided, the 99% confidence interval for the difference between the population proportions p1​−p2​ is −0.162<p<0.202, which indicates that we are 99% confident that the true difference between population proportions is contained by the interval (−0.162,0.202)

b)

If Jeff’s quarter were to truly land heads more often than his nickel, that would mean the difference p1-p2>0, which would imply that the confidence interval above should have only contained positive values.

c)

To obtain a narrower confidence interval, Jeff should decrease his confidence interval from 99% to say 95% or 90%. In other words increasing level of significance from 1% would narrow the interval.

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