Chapter 8.1, Problem 12E

(a)

To determine

Explain the margin of error when the confidence level is 95%.

Answer to Problem 12E

The margin of error is 3% when the confidence level of 95% for all sample.

Explanation of Solution

Given information:

Sample proportion = 59%

Confidence level = 95%

Margin of error = 3%

The margin of error is a statistic measurement that gives the amount of the random sampling error in the result of a survey. Higher the margin of error, the less confidence level would reflect the result of a survey of the entire population.

Hence, the margin of error is 3% for the confidence level of 95% for all sample.

(b)

To determine

To calculate: the true population proportion contain by the confidence interval.

Answer to Problem 12E

The true population proportion contain by the confidence interval is from 0.63 to 0.69.

Explanation of Solution

Given information:

Sample proportion = 59%

Confidence level = 95%

Margin of error = 3%

Sample proportion=

  $\stackrel{⌢}{p}=59\%$

Convert percentage into decimal

  $\begin{array}{l}\stackrel{⌢}{p}=59\%\\ =0.59\end{array}$

Margin of error= $E=3\%$

Convert percentage into decimal

  $\begin{array}{l}E=3\%\\ =0.03\end{array}$

The sample proportion can vary by the amount margin of error. Thus, the confidence that contain true proportion also vary from minimum to maximum sample proportion.

  $\begin{array}{l}\stackrel{⌢}{p}-E<p<\stackrel{⌢}{p}+E\\ 0.59-0.03<p<0.59+0.03\\ 0.56<p<0.62\end{array}$

Hence,The amount of confidence interval that contain the true population proportion is from 0.56 to 0.62.

(c)

To determine

Explain the confidence level.

Answer to Problem 12E

95% 0f the confidence interval has the true population proportion.

Explanation of Solution

Given information:

Sample proportion = 59%

Confidence level = 95%

Margin of error = 3%

Confidence level is the probability that the value of a parameter falls within a specified range of values. For true population proportion is from 0.56 to 0.62 the confidence level is 95%.

Hence, the confidence interval has the true population proportion for all sample is 95%.