Constructing and Interpreting Confidence Intervals. In Exercis...

Question

Constructing and Interpreting Confidence Intervals. In Exercises 13–16, use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion p; (b) identify the value of the margin of error E; (c) construct the confidence interval; (d) write a statement that correctly interprets the confidence interval.

Tennis Challenges In a recent U.S. Open tennis tournament, men playing singles matches used challenges on 240 calls made by the line judges. Among those challenges, 88 were found to be successful with the call overturned. Construct a 95% confidence interval for the proportion of successful challenges.

Accepted Answer

Welcome back, everyone. A company surveyed 250 employees and found that 85 of them preferred to work remotely. Construct a 95% confidence interval for the proportion of all employees who prefer to work remotely. So for this problem, we want to construct a confidence interval for the proportion, right? So confidence interval is equal to Phat, our sample proportion, and we are essentially adding or subtracting the margin of error E. So we can continue this equation and express the margin of error as the critical Z. See star multiplied by the standard error. Which is equal to. He had plus or minus the critical z value multiplied by. Now let's express our standard error as square root of. Phat multiplied by 1 minus p hat divided by the sample size m and we're going to use this formula to identify the answer. First of all, let's begin with the sample proportion Phat. We're given 85 of those who prefer to work remotely, and we're going to divide this number by the total number of employees surveyed 250. This gives us 0.34, so we have one of the required data points. We can state that it is 0.34. Now, what is the critical Z score? For a 95% confidence, the critical Z score is 1.960, and now our sample size M is 250. So we can see that the confidence interval is 0.34 plus or minus 1.960 multiplied by square root of. P 0.34 multiplied by 1 minus 0.34 divided by N which is 250. And now we're simply going to simplify. The part that is added or subtracted our margin of error. This gives us 0.34 plus or minus 0.0587. So now let's identify our confidence interval. It consists of the lower bound A and the upper bound B. The lower bound is simply 0.34 minus the margin of error, which is 0.0587. Rounded to two decimal places, we get 0.28, and now our upper bound B is 0.34 plus the margin of error, 0.0587. Round its two decimal places we get 0.40. So now our confidence interval. Corresponds to an interval from 0.28 to 0.40. In an inequality form, we can say that 0.28 is less than Phat, and Phat is less than 0.40. That would be our final answer and thank you for watching.

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Key Concepts

Point Estimate

Margin of Error

Confidence Interval

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