Chapter PV, Problem 32RE

(a)

To determine

To create and interpret a $95\%$ confidence interval.

Answer to Problem 32RE

The confidence interval is: $(0.11621,0.16389)$ and we conclude that we are $95\%$ confident that between $11.6\%\text{ and 16}\text{.4\%}$ of kids wear helmets.

Explanation of Solution

It is given in the question that observers in Texas watched children at play in eight communities. Let us check the conditions first as,

Random condition: It is satisfied as children are randomly selected from the population.

Independent condition: It is satisfied as we can assume that the sample is independent.

  $10\%$ condition: It is satisfied as children are less than $10\%$ of all children.

Success/failure condition: It is satisfied because both are greater than ten as,

  $\begin{array}{l}np=114>10\\ n(1-p)=700>10\end{array}$

Thus, all the conditions are satisfied. Therefore, now, to find the confidence interval we will use the calculator $\text{TI}-89$ . So, we have to first go to the $\text{Ints}$ menu and select $5:1-\text{PropZInt}$ . Then we have to enter the number of successes observed and the sample size. Then specify a confidence level. Then the interval will be calculated. Thus, we have,

  $\begin{array}{l}x=114\\ n=814\\ c=0.95=95\%\end{array}$

Thus, the confidence interval will be as:

  $(0.11621,0.16389)$ .

Thus, we conclude that we are $95\%$ confident that between $11.6\%\text{ and 16}\text{.4\%}$ of kids wear helmets.

(b)

To determine

To explain what concerns do you have about this study that might make your confidence interval unreliable.

Explanation of Solution

It is given in the question that observers in Texas watched children at play in eight communities. Thus, the confidence interval will be as:

  $(0.11621,0.16389)$ .

Thus, we conclude that we are $95\%$ confident that between $11.6\%\text{ and 16}\text{.4\%}$ of kids wear helmets. Thus, from this study we can say that the concerns we have about this study that might make your confidence interval unreliable is that the study conducted only in Texas.

(c)

To determine

To find out how many children must we observe.

Answer to Problem 32RE

  $409\text{ children}$ we must observe.

Explanation of Solution

It is given in the question that observers in Texas watched children at play in eight communities. Thus, the confidence interval will be as:

  $(0.11621,0.16389)$ .

Thus, we conclude that we are $95\%$ confident that between $11.6\%\text{ and 16}\text{.4\%}$ of kids wear helmets. It is given that margin of error is $\pm 4\%$ with $98\%$ confidence interval. Thus, the sample size is calculated as:

  $\begin{array}{l}ME=z\times \sqrt{\frac{p(1-p)}{n}}\\ \Rightarrow 0.04=2.33\times \sqrt{\frac{0.14(0.86)}{n}}\\ \Rightarrow \frac{{0.04}^2}{{2.33}^2}=\frac{0.14(0.86)}{n}\\ \Rightarrow n=409\text{ children}\end{array}$