Let P Be the Success Probability of a Bernoulli Random $\hat { p }$

question 26

Essay

Let p be the success probability of a Bernoulli random variable Y, i.e., p = Pr(Y = 1). It can be shown that $\hat { p }$ , the fraction of successes in a sample, is asymptotically distributed N(p, $\frac { p ( 1 - p ) } { n }$ 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×SE( $\hat { p }$ ))

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New Deal Era

A period in the United States during the 1930s when President Franklin D. Roosevelt's administration implemented a series of programs and policies aimed at recovering from the Great Depression.

Craft Unions

Organizations of skilled workers in the same trade or craft aiming to negotiate collectively for better wages, working conditions, and hours.

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Unions that organize workers across all crafts and trades within a single industry.

New Deal Era

A period in American history during the 1930s in which President Franklin D. Roosevelt implemented a series of economic interventions designed to recover from the Great Depression, including job creation programs, financial regulations, and social welfare initiatives.