Chapter 8.2, Problem 47E

(a)

To determine

To calculate: The confidence level for the sample in the survey.

Answer to Problem 47E

The confidence level is95%.

Explanation of Solution

Given information:

Margin of error $E=3\%$

Sample proportion $\stackrel{⌢}{p}=64\%$

Sample size $n=1028$

Formula used:

Margin of error $E=z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\times \sqrt{\frac{\stackrel{⌢}{p}\left(1-\stackrel{⌢}{p}\right)}{n}}$

Calculation:

Sample proportion $\text{<mover accent="true"><mo>p</mo><mo>⌢</mo></mover>}\text{is}\text{64\%}$ .

Convert 64% into decimal.

  $\frac{64}{100}=0.64$

Now, calculate the margin of error. Use the formula $E=z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\times \sqrt{\frac{\stackrel{⌢}{p}\left(1-\stackrel{⌢}{p}\right)}{n}}$ .

  $\begin{array}{c}E=z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\times \sqrt{\frac{\stackrel{⌢}{p}\left(1-\stackrel{⌢}{p}\right)}{n}}\\ =z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\times \sqrt{\frac{0.64\left(1-0.64\right)}{1028}}\\ =0.01497z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}$........(1)

Margin of error is 3%.

Convert 3% into decimal.

  $\frac{3}{100}=0.03$

Substitute 0.03 for E in equation (1).

  $\begin{array}{c}0.03=0.01497z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{0.03}{0.01497}\\ \approx 2\end{array}$

Calculate theconfidence level, use table A.

  $\begin{array}{c}P\left(-2<Z<2\right)=P\left(Z<2\right)-P\left(Z<-2\right)\\ =0.9772-0.0228\\ =0.95\end{array}$

Thus the confidence level is 0.95.

Convert 0.95 into percentage.

  $0.95\times 100=95$

Hence, the required confidence level is 95%.

(b)

To determine

Explain the reason that which of the practical difficulty lead to biased results for this survey.

Answer to Problem 47E

Nonresponse bias can lead to biased results for this survey.

Explanation of Solution

There are three practical difficulties that can lead to biased result in any survey. These are selection bias, response bias and nonresponse bias.

In this survey most teens will not participate in survey.

Hence, nonresponse bias can lead to biased results for this survey.