Chapter 10, Problem 2CRE

(a)

To determine

To Explain: that this is an experiment study or observational study.

Answer to Problem 2CRE

Observational study

Explanation of Solution

Given:

  

Experiment Study

In order to study their reactions, an experiment intentionally forces certain treatment upon individuals.

Observational Study

Without disrupting the scene, they are studying, an observational study attempts to collect details.

Therefore, Observational study

(b)

To determine

To Construct: and interpret a 95 % confidence interval for the difference in the proportion of female Hispanic drivers who wear seat belts in the two cities.

Answer to Problem 2CRE

(0.149, 0.353)

Explanation of Solution

Given:

  

Formula used:

For the confidence interval

  $\left({\widehat{p}}_1-{\widehat{p}}_2\right)\pm z_{\alpha /2}\times \sqrt{\frac{{\widehat{p}}_1\left(1-{\widehat{p}}_1\right)}{n_1}+\frac{{\widehat{p}}_2\left(1-{\widehat{p}}_2\right)}{n_2}}$

Calculation:

The sample proportions are

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}=\frac{183}{220}=0.832\\ {\widehat{p}}_2=\frac{x_2}{n_2}=\frac{68}{117}=0.581\end{array}$

For the confidence level $1-\alpha =0.95$ , determine $z_{\alpha /2}=z_{0.025}$

the z-score is

  $z_{\alpha /2}=1.96$

The confidence interval are

  $\begin{array}{l}\left({\widehat{p}}_1-{\widehat{p}}_2\right)-z_{\alpha /2}\times \sqrt{\frac{{\widehat{p}}_1\left(1-{\widehat{p}}_1\right)}{n_1}+\frac{{\widehat{p}}_2\left(1-{\widehat{p}}_2\right)}{n_2}}\\ =\left(0.832-0.581\right)-1.96\sqrt{\frac{0.832\left(1-0.832\right)}{220}+\frac{0.581\left(1-0.581\right)}{117}}\\ =0.149\\ \left({\widehat{p}}_1-{\widehat{p}}_2\right)+z_{\alpha /2}\times \sqrt{\frac{{\widehat{p}}_1\left(1-{\widehat{p}}_1\right)}{n_1}+\frac{{\widehat{p}}_2\left(1-{\widehat{p}}_2\right)}{n_2}}\\ =\left(0.832-0.581\right)+1.96\sqrt{\frac{0.832\left(1-0.832\right)}{220}+\frac{0.581\left(1-0.581\right)}{117}}\\ =0.353\end{array}$

95% are sure that the percentage of Hispanic female drivers in New York is between 0.149 and 0.353 higher than the proportion of Hispanic female drivers in Boston.

(c)

To determine

To explain: the confidence interval in part (b) provides strong evidence that this is valid for Hispanic female drivers, to support the response.

Answer to Problem 2CRE

Yes

Explanation of Solution

Given:

  

From part (b)

(0.149, 0.353)

In the absence of a confidence interval of 0, the interval suggests that a lower proportion of drivers in Boston are wearing seat belts than in New York and that there is ample evidence supporting the claim.