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.

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Construct a 95% confidence interval for the proportion of medical malpractice lawsuits that are dropped or dismissed.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. In a poll of 1500 registered voters, 1,035 said that they planned to vote in the upcoming election. Construct a 95% confidence interval for the proportion of all registered voters who plan to vote. Write a statement that correctly interprets the confidence interval. Awesome. So it appears for this particular prompt we're asked to solve for two separate answers. Our first answers we're asked to create or construct a 95% confidence interval for the proportion of all registered voters who plan to vote. And our second answer we're trying to solve for is we're asked to create or write a statement that correctly interprets our confidence interval value that we find that we're about to find together. So now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to write down all of our known variables, all of our variables that are given to us by the problem itself. So we're told that the sample size, which we'll call N, we're going to denote sample sizes N, is equal to 1500. We're also told the number of voters who plan to vote, we'll call this value X and X is equal to 10, so 1,035. Awesome. So once again, our sample size is equal to 1500 and our number of voters who plan to vote in the upcoming election is X is equal to 1,035. Awesome. We also can determine the sample proportion, which we'll call this value Phat, and the sample proportion P is equal to X divided by N, which is equal to 1,035 divided by 1500, which is going to be equal to 0.69 when we plug that into our calculator. We're also told that the confidence interval that we're trying to solve for is 95%. And for a 95% confidence interval, the critical value, which is denoted as Z subscript alpha divided by 2, and this value is going to be equal to 1.96. Fantastic. So now, Our first step, or rather our second major step, or we could say first step towards actually solving the problem, we can, we now need to determine the best point estimate of P. So let's say that Phat, as we determine, so Phat is equal to 0.69. So, since we determine our Phat value to be equal to 0.69, our second step is we need to determine the margin of error. So, as we should recall, the margin of error equation states that E, the margin of error, is equal to Z subscript alpha divided by 2, multiplied by the square root of Phat multiplied by parentheses 1 minus Phat all divided by N. And when we plug in all of our known variables, we will find that this is equal to 1.96 multiplied by the square root of 0.69 multiplied by 1 minus 0.69. All divided by 1500. And when we plug that into our calculator, we will find that E, our margin of error, so we'll determine that E. Our margin of error is going to be equal to 0.0234. Fantastic. So now our third step is we need to construct a 99% confidence interval. So our confidence interval CI will be equal to Phat plus or minus E, or margin of error, which in this case, is going to be equal to when you plug in unknown variables 0.69 plus or minus 0.0234. So that would mean that our confidence interval for the conditions that we're trying to solve for is going to be equal to parentheses 0.667.0.713. Fantastic. And that is our confidence interval value that we were ultimately trying to solve for. Hooray, we did it. So, that will mean that our next step now is we need to determine our interpretation. So we need to write a statement that correctly interprets our confidence interval. So, therefore, in conclusion, we are 95% confident that the true proportion of all registered voters who plan to vote in the upcoming election is between, note 66.7% and 71.3%. So that will mean when we summarize our final Data points here, and we write our final answer, that will mean that our final answer, without a doubt, or rather answers will have to be the following. So once again, our confidence interval is parentheses 0.667.0.713, and as for our interpretation statement, we are 95% confident that the true proportion of all registered voters who plan to vote in the upcoming election is between 66.7% and 71.3%. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Point Estimate

Margin of Error

Confidence Interval

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