Chapter 18, Problem 5E

a.

To determine

Obtain the margin of error for the proportion of all adult Wisconsin residents thinking that things are going in the wrong direction for 90% confidence.

Answer to Problem 5E

The margin of error for the proportion of all adult Wisconsin residents who think things are going in the wrong direction for 90% confidence is 0.039 or 3.9%.

Explanation of Solution

Calculation:

The survey conducted by St. Norbert’s College in Green Bay, Wisconsin and Wisconsin Public Radio on the sample of 402 Wisconsin resident adults, 66% said that things in the country are going in the wrong direction.

Margin of error:

The margin of error for the estimate of proportion, p is,

$ME=z*\times \sqrt{\frac{\widehat{p}\widehat{q}}{n}}$

Where, $z*$ denotes critical value from the standard normal distribution, n be the number of samples, $\widehat{p}$ be the sample proportion and $\widehat{q}=1-\widehat{p}$.

Critical value for 90% confidence:

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0.0 and standard deviation as 1.0.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tail for the region of the curve to shade.
• Enter the Probability as 0.10.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the critical values at 90% confidence are $z*=\pm 1.645$.

The margin of error is:

Substitute z as 1.645, $\widehat{p}$ as 0.66 and $\widehat{q}$ as 0.34 $\left(=1-\widehat{p}\right)$ and n as 402 in the formula,

$\begin{array}{c}ME=1.645\times \sqrt{\frac{\left(0.66\right)\left(0.34\right)}{402}}\\ =1.645\times \sqrt{\frac{0.2244}{402}}\\ =1.645\times 0.02363\\ \approx 0.039\end{array}$

Thus, the margin of error for the proportion of all adult Wisconsin residents who think things are going in the wrong direction for 90% confidence is 0.039 or 3.9%.

b.

To determine

Explain whether the margin of error is larger or smaller for 95% confidence.

Answer to Problem 5E

The margin of error is larger for 95% confidence.

Explanation of Solution

Justification:

Critical value for 95% confidence:

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 0.0 and standard deviation as 1.0.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tail for the region of the curve to shade.
• Enter the Probability as 0.05.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the critical values at 95% confidence are $z*=\pm 1.96$.

For this particular problem, the values of n, $\widehat{p}$ and $\widehat{q}$ are constant. As a result, the margin of error depends only upon the critical value $z*$. The margin of error increases with the increase in $z*$.

The critical value for 95% confidence is 1.96 which is greater than the critical value 1.645 of 90% confidence.

Therefore, the margin of error for 95% confidence is larger than that obtained for the 90% confidence.