Constructing Confidence Intervals In Exercises 11 and 12, cons...

Question

Constructing Confidence Intervals In Exercises 13 and 14, construct a 99% confidence interval for the population proportion. Interpret the results.

New Year’s Resolutions In a survey of 1790 U.S. adults in a recent year, 816 have a New Year’s resolution related to their health. (Adapted from Finder)

Accepted Answer

Hello. In this video, out of 1250 randomly selected registered voters, 615 said they plan to vote in the upcoming local election. We want to construct a 90% and a 95% confidence interval for the proportion of all registered voters who plan to vote, and then we want to determine which interval is wider. OK, so let's just go ahead and list out some given information from the problem. First, we are working with a random with a random sample of 1250 people. So we are going to let N equal to 1250. Now, the number of voters that we're interested in, which are the ones that plan to vote in the next local election, is 615. So, what we need to do in order to construct these confidence intervals is we first need to define our point estimate. Now, the point estimate is the sample of people that we are interested in in the total amount of people we interviewed. Now, in this case here, since we are interested in the 615. People who plan to vote out of the total sample size, our point estimate is defined as 615 divided by 1250, and this is going to give us an approximate value of 0.492. Now, the next thing we want to do is we want to go ahead and define our standard error. Now, the standard error is going to be the square root. Of the point estimate multiplied by its complement, so 1 minus the point estimate and divide this value by the total sample size. This is going to be the square root. Of 0.492, multiplied by 1 minus 0.492, and this will be divided by the sample size, which is 1250. And this is going to give us an approximate standard error of 0.0141. Now, what we need to do is we need to determine if we can apply a normal distribution. Since our sample size is fairly large with 1250 people, and our point estimate is nowhere near 0 or 1, we can apply a Z normal distribution. So the next thing we want to do is we want to go ahead and determine the Z scores. Determine the Z scores for both the 90% and 95% confidence intervals. So, starting with the 90% confidence interval, we want to find the Z score. Which is defined or the Z score with respect to the 90% confidence interval. By using a Z table, we get a Z score of 1.645. For the 95% confidence interval. We want to find the confidence or the Z score for the 95%, and by using a Z table, we get a Z value of 1.96. Now that we have the Z scores, we want to go ahead and calculate the margin of error for both of the confidence intervals. Now, the margin of errors are defined as the standard error multiplied by the Z scores we calculated. So, starting with our 90% confidence interval. The margin of error for this confidence interval is going to be the standard error, which is 0.0141, multiplied by that Z score, which is 1 point. 645, and that is going to give us a margin of error for the 90% confidence interval of approximately 0.0232. And we are going to repeat this process for the 95% confidence interval. That is going to be the standard error, which is 0.0141, multiplied by that Z score, which is 1.96, and that is going to give us a margin of error of approximately 0.0276. Now that we've calculated the margin of errors and we have our point estimate, we can now construct the confidence intervals. Now, the confidence intervals, or at least the end points of them, are defined as our point estimate, plus or minus our margin of error. So, starting with our 90% confidence interval, those end points of the interval are going to be defined as the point estimate, which is 0.492, plus or minus that margin of error, which is 0.0232. By adding and subtracting these values together, the end points of this of this confidence interval is going to be 0.4688 for the lower boundary, and the upper interval or upper boundary of 0.5152. This is going to be the confidence interval for the 90% confidence level. And we will go ahead and repeat this process for the 95% confidence interval. So again, that is our point estimate, which is 0.492 plus or minus that margin of error, which is 0.0276. And by adding them subtracting these values together, we get a lower bound of the interval of 0.4644. And an upper boundary of 0.5196. And that is going to be the confidence interval for the 95% confidence level. Now, in determining which one of these is wider, the 95% confidence interval is going to be the wider interval. And with that being said, the solution to this problem is going to be a. So, I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Confidence Interval

Population Proportion

Width of Confidence Intervals

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