Chapter 10, Problem 9CRE

(a)

To determine

To Explain: that this is an observation study or an experiment study yes or not.

Answer to Problem 9CRE

Experiment

Explanation of Solution

An experiment intentionally applies a certain procedure on people to study their reactions. An observational analysis aims to gain details without disrupting the scene they are watching.

(b)

To determine

To Explain: on the basis of data, can it infer that leaving a message raises the proportion of households interviewed in the end. Provide sufficient supporting evidence for the responses.

Answer to Problem 9CRE

Yes

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=200\\ n_1=200+91=291\\ x_2=58\\ n_2=58+42=100\end{array}$

Formula used:

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ {\widehat{p}}_2=\frac{x_2}{n_2}\end{array}$

  ${\widehat{p}}_p=\frac{x_1+x_2}{n_1+n_2}$

  $z=\frac{{\widehat{p}}_1-{\widehat{p}}_2}{\sqrt{{\widehat{p}}_p\left(1-{\widehat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

Calculation:

The hypotheses are

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_0:p_1>p_2\end{array}$

The sample proportion is

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}=\frac{200}{291}=0.6873\\ {\widehat{p}}_2=\frac{x_2}{n_2}=\frac{58}{100}=0.58\end{array}$

  $\begin{array}{c}{\widehat{p}}_p=\frac{x_1+x_2}{n_1+n_2}\\ =\frac{200+58}{291+100}\\ =\frac{258}{391}\\ =0.6598\end{array}$

The test statistic is

  $z=\frac{{\widehat{p}}_1-{\widehat{p}}_2}{\sqrt{{\widehat{p}}_p\left(1-{\widehat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

  $\begin{array}{c}=\frac{0.6873-0.58}{\sqrt{0.6598\left(1-0.6598\right)\sqrt{\frac{1}{291}+\frac{1}{100}}}}\\ =1.95\end{array}$

The P-value is the chance of having the value of the test numbers, or a more drastic value.

  $\begin{array}{c}P=P\left(Z>1.95\right)\\ =P\left(Z<-1.95\right)\\ =0.0256\end{array}$

Reject the null hypothesis if the P-value is lower than the significance level:

  $P<0.05\Rightarrow \text{Raject }{H}_0$