Chapter 19, Problem 27E

(a)

To determine

To create a $90\%$ confidence interval for the percentage of deer that may carry such ticks.

Answer to Problem 27E

  $(0.155,0.263)$ .

Explanation of Solution

It is given in the question that wildlife biologists inspect deer taken by hunters and find them carrying ticks that test positive for Lyme disease. Thus, we have,

Let us first check the conditions for the following as:

Random condition: It is satisfied as it is assumed that the sample is randomly selected.

Normal condition: It is satisfied as,

  $\begin{array}{l}np=32>10\\ n(1-p)=121>10\end{array}$

$10\%$ condition: It is satisfied as the sample size is less than $10\%$ of the population size.

Thus, all the conditions are met. 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,

  $\begin{array}{l}x=32\\ n=153\\ c=0.90\end{array}$

Thus, the confidence interval will be as:

  $(0.155,0.263)$

(b)

To determine

To find out how many deer must they inspect.

Answer to Problem 27E

  $611$ .

Explanation of Solution

It is given in the question that wildlife biologists inspect deer taken by hunters and find them carrying ticks that test positive for Lyme disease. Thus, the confidence interval will be as:

  $(0.155,0.263)$

Thus, to find the deer they must inspect we need to find the margin of error as:

  $\begin{array}{l}ME=1.645\times \sqrt{\frac{0.209\times 0.791}{153}}\\ =0.054\\ M{E}_N=\frac{0.054}{2}\\ =0.027\\ \text{Thus, }0.027=1.645\times \sqrt{\frac{0.209\times 0.791}{n}}\\ \text{Solve for }n\\ \Rightarrow 0.0164=\sqrt{\frac{0.209\times 0.791}{n}}\\ \Rightarrow n=611\end{array}$

Thus, the number of deer they must inspect will be $611$ .

(c)

To determine

To explain what concerns do you have about this sample.

Explanation of Solution

It is given in the question that wildlife biologists inspect deer taken by hunters and find them carrying ticks that test positive for Lyme disease. Thus, the confidence interval will be as:

  $(0.155,0.263)$

Thus, the sample may not be random or representative. Deer that may legally be hunted may not be representative of all sexes and ages.