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?

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?

Key Concepts

Point Estimate

The point estimate is a single numerical value used to estimate an unknown parameter from a population. In survey research, it is often computed as the sample proportion and serves as the most likely value for the parameter of interest.

Margin of Error

The margin of error reflects the extent of the uncertainty associated with the point estimate. It is calculated by multiplying the appropriate critical value (based on the desired confidence level) by the standard error, indicating how much the sample estimate may deviate from the true population parameter.

Confidence Interval

A confidence interval is a range of values derived from the sample data that is believed, with a certain degree of confidence, to contain the true population parameter. It is expressed as the point estimate plus and minus the margin of error.

Confidence Level

The confidence level represents the percentage of times that the confidence interval, computed from repeated random samples, would encompass the true population parameter. It is a measure of the reliability of the estimation procedure, with common levels being 95% or 75% as seen in the problem.

Effect of Sample Size

Sample size plays a critical role in statistical estimation; larger sample sizes tend to produce smaller standard errors, resulting in narrower confidence intervals. This means that increasing the sample size improves the precision of the point estimate and overall reliability of the conclusions drawn from the interval.

Transcript

00:01 This problem refers to a survey of 1 ,000 adults, and it was concluded with 95 % confidence that from 49 % to 55 % of americans believe that big -time college sports programs corrupt the process of your higher education institutions.

00:19 So n was 1 ,000, and we were given a confidence interval at the 95 % confidence level of 0 .49.

00:32 Up to .55.

00:36 So if we think of that in terms of a number line, the low end is 49%, the high end is 55%.

00:45 And part a is asking you to calculate the point estimate, and the point estimate is identified with the variable p prime, and every time you do a confidence interval, the point estimate is found right in the middle of your confidence interval, so in order to find p prime, we could average the two end points of our confidence interval.

01:10 So we'll take .49 plus .55 and we'll divide by 2.

01:18 And in doing so, you're going to get a point estimate of 52 % or .52.

01:25 The other part of this problem or part a asked you to find the error bound and the error bound of the proportion is going to be that wiggle room.

01:38 So it's the distance from the center to each end point of the confidence interval.

01:46 So we can get that by taking the high end of the confidence interval minus the point estimate, which gets you 0 .03.

01:58 Or you can go from the point estimate, 0 .52, and subtract the low end.

02:05 And either way, we get an error bound of 0 .03.

02:11 In part b of this problem, it's asking, can we, with this 95 % confidence, conclude that more than half of all american adults believe this? and the answer is no, we cannot conclude that more than half.

02:29 And the reason being would be because our confidence interval spans, from 49 % up to 55%.

02:59 So that means your true proportion can be anywhere in there, in that interval.

03:31 And since the interval goes as low as 0 .49, we might be here, which could be less than half.

03:40 So therefore, no, we cannot conclude that more than half of all american adults believe this based on this confidence interval.

03:50 As we go into part c, part c is asking you to construct a 75 % confidence interval.

04:02 So we're going to use the p -prime that we found in part a, and that was 0 .52, and we're going to use the fact that we surveyed 1 ,000 people, and we need to find the confidence interval.

04:17 Well, in order to find the confidence interval, we will need to find the error bound of that proportion, using the formula z of alpha over 2 multiplied by the square root of p prime times q prime over n.

04:33 And in order to calculate the z score associated with this alpha over 2, we will need to draw our bell -shaped curve, which then puts 75 % confidence into center, and our alpha is the part of the curve that is not accounted for.

04:53 In that confidence interval.

04:56 So there is 25 % of the curve still unaccounted for.

05:00 And because the bell -shaped curve is symmetric, each tail will have half of that or 0 .125.

05:09 So in each tail, we can put a 0 .125 or 12 .5%...

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