A national survey of 1,000 adults was conducted on May...

A national survey of 1,000 adults was conducted on May 13,2013 by Rasmussen Reports. It concluded with $95 \%$ confidence that $49 \%$ to $55 \%$ of Americans believe that big-time college sports programs corrupt the process of higher education. a. Find the point estimate and the error bound for this confidence interval. b. Can we (with $95 \%$ confidence) conclude that more than half of all American adults believe this? c. Use the point estimate from part a and $n=1,000$ to calculate a $75 \%$ confidence interval for the proportion of American adults that believe that major college sports programs corrupt higher education. d. Can we (with $75 \%$ confidence) conclude that at least half of all American adults believe this?

Transcript

00:01 All right, in your question, you're told that you had a sample size of 1 ,000.

00:04 A 95 % confidence interval was created, and it showed that between 49 % to 55 % of americans believe that big time college sports programs corrupt the process of higher education.

00:17 So that confidence interval would change to decimal at 0 .49 to 0 .55.

00:25 And from that confidence interval, you want to find the point estimate for the sample proportion or the population proportion who believe this.

00:35 Well, that's just going to be the number in the middle here.

00:38 So all we have to do in the middle of the confidence interval, all we have to do is add those two values and divide by 2.

00:47 And you would find that that value is 0 .52.

01:00 Now, the error is how much above or below those values are in the confidence interval above the estimate.

01:09 And that's going to work out to 0 .03.

01:13 So if you add the error, that's how you get to 0 .55.

01:17 If you subtract it from 0 .52, that's how you get to 0 .49.

01:22 That completes part a.

01:25 Part b says, can we conclude with 95 % confidence that more than half of american adults believe this? and you would say no based on, let's say, i'm going to say a proportion less than 0 .5 is possible in our confidence interval.

02:05 So basically, we can't say that with 95 % confidence because in our confidence interval, we go below half the population.

02:12 So because that's a possibility within that confidence interval, we say no.

02:18 Now, if the entire confidence interval would have been above 0 .5, we would have said yes based on the confidence.

02:24 All right, so now what we're trying to do is repeat this procedure or create a new confidence interval.

02:29 It's still going to have the sample size 1 ,000.

02:33 It's still going to have the same sample proportion or point estimate at 0 .52.

02:40 But we are reducing this confidence level drastically to 75%.

02:44 So to do that, what we have to first understand is we're going to build our confidence interval by taking our sample proportion, plus or minus a critical value, which is based on your confidence level, 75%, times your standard error, which is going to be p hat times 1 minus p hat divided by our sample size.

03:13 Now, the first thing we have to address here is that critical value...

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