8. Sampling Distributions & Confidence Intervals: Proportion

A factory manager wants to estimate the proportion of defective items produced. In a batch of 20 items, the factory has produced 6 with defects. Find the margin of error for a 98% confidence interval for the true proportion of defective items.

Over the first $20$ days of the semester, one student is late to class on $6$ days. Construct a $98\%$ confidence interval for the true proportion of time this student is late. 

A previous study found that your school consists of $60\%$ White/Caucasian students. You want the $98\%$ confidence interval for the proportion of White/Caucasian students to be no more than $.05$ away from the true proportion. How many students must you include in a sample to create this confidence interval? 

A labor economist wants to estimate, with $90\%$ confidence, the proportion of remote workers in the workforce. The economist wants the estimate to be accurate within $4\%$ of the true population proportion. What is the minimum sample size needed for this estimate?

Your company has asked you to estimate the proportion of people who prefer the color red over other primary colors for manufacturing purposes. If they want the estimate to be within $.01$ of the true proportion with $95\%$ confidence, how many people should you survey?  

You want to make a $97\%$ confidence interval for the population proportion of people between $20-30$ years old who have gotten a speeding ticket in the past $2$ years. A prior study found that $26\%$ of people between $20-30$ years old have received a speeding ticket in the last year. If you want your estimate to be accurate within $3\%$ of the true population proportion, what is the minimum sample size needed? 

Make a confidence interval for $p$ given the following values.

$x=314,n=500,C=99\%$

Make a confidence interval for $p$ given the following values.

$p̂=0.15,n=60,C=90\%$

An economist is evaluating how frequently the U.S. inflation rate exceeds the Federal Reserve's long-term target of 2% per $yr\approx 0.17\%$ per month. The economist finds that in 34 of the 48 sampled months, the monthly inflation rate did exceed $0.17\%$.

Make a $95\%$ confidence interval for the true proportion of months in which the inflation rate exceeds the target.

We are ___$\%$ confident that the inflation rate exceeds the target in between (_––––,_––––) of months. 

An economist is evaluating how frequently the U.S. inflation rate exceeds the Federal Reserve's long-term target of 2% per $yr\thickapprox0.17\%$ per month. The economist finds that in 34 of the 48 sampled months, the monthly inflation rate did exceed $0.17\%$.

Under stable conditions the inflation rate should not exceed the target more than $20\%$ of the time. Can the economist conclude that inflation has exceeded the target more than $20\%$?