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

Over the first twenty days of the semester, a student is late to class on six of those days. We're asked here to construct a 98% confidence interval for the true proportion of time that this student is late. Now if you already worked through the previous practice problem, we actually already calculated the margin of error. But I'm gonna go through that process really quickly again here. That was already steps one through four. So really, this problem is just putting that together in step five to actually construct our confidence interval. But I'm gonna go ahead and go back through that process really quickly going through our steps. We first need to verify that the number of successes and failures is at least five because they were late to class on six days and that's what we're considering to be a success here, that is at least five. On the other fourteen days that they were late, that is our failures, in this case that is also at least five. Moving on to step two, finding our critical value based on having a 98% confidence interval. That corresponds to a critical z value, z alpha over two, of 2.326. Then p hat, that's x over n. That was equal to six over 20, which is 0.3 dividing x by n. Then we can find our margin of error by taking that z alpha over two, two point three two six, multiplying that by the square root of 0.3 times one minus 0.3, so 0.7 over n. That's our total number of days in this case, which is 20 here. I'm just using my margin of error equation, which is right here. Now this worked out to give us a margin of error of 0.238. And from here, we are at step five where we're actually constructing our confidence interval. If you wanna go through those first four steps in more details, click back to that previous problem. Alright. Continuing on on step five here, we're finding the upper and lower bounds of that confidence interval by subtracting and adding that margin of error to our point estimate p hat. So our confidence interval here is going to be equal to 0.3 minuteus zero point two three eight and zero point three plus 0.238. This results in a confidence interval here. That lower bound is gonna give us 0.062 and our upper bound is 0.538. So this is a rather large interval here because our lower bound is really small. Our upper bound is point five three eight. So this is our full confidence interval. This is our confidence interval for the true proportion of time that the student is late. Remember, an important part of working with confidence intervals is being able to actually interpret what they mean. Based on what we found here, we can say that we are 98% confident that the true proportion of time that this student is late is between 6.253.8%. Feel free to let us know if you have any questions and I'll see you in the next one.