L Let $p$ Be the Success Probability of a Bernoulli Random Variable

question 54

Essay

L Let $p$ be the success probability of a Bernoulli random variable $Y$ , i.e., $p = \operatorname { Pr } ( Y = 1 )$ . It can be shown that $\hat { p }$ , the fraction of successes in a sample, is asymptotically distributed $N \left( p , \frac { p ( 1 - p ) } { n } \right)$ . Using the estimator of the variance of $\hat { p } , \frac { \hat { p } ( 1 - \hat { p } ) } { n }$ , construct a $95 \%$ confidence interval for $p$ . Show that the margin for sampling error simplifies to $1 / \sqrt { n }$ if you used 2 instead of $1.96$ assuming, conservatively, that the standard error is at its maximum. Construct a table indicating the sample size needed to generate a margin of sampling error of $1 \% , 2 \% , 5 \%$ and $10 \%$ . What do you notice about the increase in sample size needed to halve the margin of error? (The margin of sampling error is $1.96 \times S E ( \hat { p } )$ .

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