Chapter 22, Problem 17E

(a)

To determine

To find: the sort of design used by researchers.

Answer to Problem 17E

Prospective observational study

Explanation of Solution

It is a prospective study of observation. A prospective research defines and observes a group of individuals in the future.

(b)

To determine

To Write: the suitable hypotheses.

Answer to Problem 17E

  $\begin{array}{c}{\text{H}}_0:p_1-p_2=0\\ {\text{H}}_1:p_1-p_2\ne 0\end{array}$

Explanation of Solution

Here want to examine whether parental altitude affects the smoking choices of teenagers. Suppose $p_1$ represent the percentage of smoking reported by students with disapproving parents and $p_2$ represent the percentage of smoking started by students with approving parents. The parental mindset affects the choices of teenagers regarding smoking, so $p_1$ should not be equal to $p_2$ . So, our hypotheses are going to be

  $\begin{array}{c}{\text{H}}_0:p_1-p_2=0\\ {\text{H}}_1:p_1-p_2\ne 0\end{array}$

(c)

To determine

To explain: to satisfy the assumptions and conditions required for conclusion.

Explanation of Solution

It is believed that samples have been arbitrary selected. The total sample size would be less than 10% of all learners. It is fair to assume that both groups are independent, since the samples were selected at random. Among the 284 students who suggested that their parents rejected of children's smoking, the number of successes is 54 and the number of failures is 284-54= 230 and among the 41 students who initial said their parents were lenient about smoking, the number of successes is 11 and the number of failures is 41-11= 30. These numbers, for each group, are at minimum 10.

The assumptions and conditions required for inference are met.

(d)

To determine

To Test: the hypothesis and state the conclusion.

Explanation of Solution

Given:

  $n_1=284$

  $n_2=41$

  $y_1=54$

  $y_2=11$

Formula used:

  $\begin{array}{c}{\widehat{p}}_{Pooled}=\frac{y_1+y_2}{n_1+n_2}\\ S{E}_{Pooled}\left({\widehat{p}}_1-{\widehat{p}}_2\right)=\sqrt{\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_1}+\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_2}}\end{array}$

Calculation:

  $\begin{array}{l}{\widehat{p}}_1=\frac{54}{284}=0.1901\\ {\widehat{p}}_2=\frac{11}{41}=0.2683\end{array}$

Pool the sample data.

  $\begin{array}{c}{\widehat{p}}_{Pooled}=\frac{y_1+y_2}{n_1+n_2}\\ =\frac{54+11}{284+41}\\ =\frac{65}{325}\\ =0.2\end{array}$

Estimate $SD\left(p_1-p_2\right)$ . Therefore,

  $\begin{array}{c}S{E}_{Pooled}\left({\widehat{p}}_1-{\widehat{p}}_2\right)=\sqrt{\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_1}+\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_2}}\\ =\sqrt{\frac{0.2\left(1-0.2\right)}{284}+\frac{0.2\left(1-0.2\right)}{41}}\\ =\sqrt{\frac{0.2\left(0.8\right)}{284}+\frac{0.2\left(0.8\right)}{41}}\\ =0.0668\end{array}$

Observed difference

  $\begin{array}{c}{\widehat{p}}_1-{\widehat{p}}_2=0.1901-0.2683\\ =-0.0782\end{array}$

The z-score is

  $\begin{array}{c}z=\frac{\left({\widehat{p}}_1-{\widehat{p}}_2\right)-0}{S{E}_{pooled}\left({\widehat{p}}_1-{\widehat{p}}_2\right)}\\ =\frac{-0.0782-0}{0.0668}\\ =-1.17\end{array}$

The P-value is

  $\begin{array}{c}\text{P-value = 2}p\left(z<-1.17\right)\\ =2\left(0.1211\right)\\ =0.2422\end{array}$

To consider the null hypothesis, the P-value = 0.2422 is high enough. It may assume that the behaviour of parents does not affect the decisions of teenagers regarding smoking.

(e)

To determine

To Explain: The P-value means.

Explanation of Solution

The P-value is the likelihood that a test statistic will be as large as or greater than the observed value if the null hypothesis is valid. So, if the P-value is 0.2422, this means that there is a 24.22 percent probability that natural variance in sampling will produce a 7 percent difference in sample proportion when this difference is actually 0.0.

(f)

To determine

To find: the type of error committed if the statement is actually incorrect.

Answer to Problem 17E

Type II

Explanation of Solution

There have introduced the null hypothesis here. If the inference is simply false, this suggests supporting the null hypothesis although it is incorrect. So, we made errors with Type II.