Question

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

Short answer

a. The point estimate and the error bound for this confidence interval are$0.52$and$0.03$respectively.

b. No

c. $75\%$confidence interval for the proportion of American adults that believe that major college sports programs corrupt higher education is $(0.502,0.538)$

d. Yes

Step-by-step solution

Part (a) Step 1: Given Information

A national survey of$1000$adults was conducted.

Per cent of Americans that believe that big-time college sports programs corrupt the process of higher education = $49\%\text{to}55\%$

Confidence level =$95\%$

Part (a) Step 2: Explanation

The confidence interval = $(0.49,0.55)$

Calculating the error bound = $\frac{\text{Upper limit-lower limit}}{2}=\frac{0.55-0.49}{2}=0.03$

Calculating the point estimate =$\frac{\text{Upper limit+lower limit}}{2}=\frac{0.55+0.49}{2}=0.52$

Part (b) Step 1: Given Information

A national survey of$1000$adults was conducted.

Per cent of Americans that believe that big-time college sports programs corrupt the process of higher education =$49\%-55\%$

Confidence level =$95\%$

Part (b) Step 2: Explanation

As the confidence interval includes values$\le 0.50$we can not conclude that more than half of all American adults believe this.

Part (c) Step 1: Given Information

A national survey of $1000$adults was conducted.

Per cent of Americans that believe that big-time college sports programs corrupt the process of higher education =$49\%-55\%$

Confidence level =$95\%$

Part (c) Step 2: Explanation

Given,

n= $100$and confidence interval= $75\%$

$\Rightarrow \alpha =1-0.75=0.25$

Now, Z= $1.15$and point estimate =$0.52$

Calculating the error bound = $1.150\sqrt{\frac{0.52\times 0.48}{1000}}=0.018$

Confidence interval = point estimate $\pm$error bound$\Rightarrow (0.52\pm 0.018)$

Hence the confidence interval = $(0.502,0.538)$

Part (d) Step 1: Given Information

A national survey of$1000$ adults was conducted.

Per cent of Americans that believe that big-time college sports programs corrupt the process of higher education =$49\%-55\%$

Confidence level =$95\%$

Part (d) Step 2: Explanation

Yes, as the confidence interval does not lie below $0.50$we can conclude that at least half of all American adults believe this.