Chapter 19, Problem 23E

(a)

To determine

To find a $98\%$ confidence interval.

Answer to Problem 23E

  $(0.182,0.218)$ .

Explanation of Solution

It is given in the question that vitamin D whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong healthy bones. The bone disease rickets was largely eliminated. 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,

  $\begin{array}{l}np=540>10\\ n(1-p)=2160>10\end{array}$

$10\%$ 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,

  $\begin{array}{l}x=540\\ n=2700\\ c=0.98\end{array}$

Thus, the confidence interval will be as:

  $(0.182,0.218)$

(b)

To determine

To explain carefully what your interval means.

Explanation of Solution

It is given in the question that vitamin D whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong healthy bones. The bone disease rickets was largely eliminated. Thus, the confidence interval will be as:

  $(0.182,0.218)$

Thus, our confidence interval means that we are $98\%$ confident that the interval $(0.182,0.218)$ contains the true population proportion of English children who have a vitamin D deficiency.

(c)

To determine

To explain what “ $98\%$ confidence” means.

Explanation of Solution

It is given in the question that vitamin D whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong healthy bones. The bone disease rickets was largely eliminated. Thus, the confidence interval will be as:

  $(0.182,0.218)$

Thus, “ $98\%$ confidence” means that about $98\%$ of all random samples will produce a confidence interval that contains the true proportion.