Chapter 18, Problem 13E

(a)

To determine

To explain can you apply the central limit theorem to describe the sampling distribution model for the sample proportion of children who are near-sighted and also check the condition.

Answer to Problem 13E

Yes, we can apply the central limit theorem to describe the sampling distribution model for the sample proportion of children who are near-sighted.

Explanation of Solution

It is given in the question that it is generally believed that near-sightedness affects about $12\%$ of all students. And a school district has registered $170$ incoming kindergarten children. Thus, we have,

  $\begin{array}{l}\mu =p=0.12\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.12(0.88)}{170}}\\ =0.025\end{array}$

Therefore, let us check the conditions as:

Randomization condition: It is satisfied as we assume that the $170$ children are a representative sample of all children.

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

Success/failure condition: As, $np=20.4,n(1-p)=149.6$ both are greater than ten then it is also satisfied.

Thus, all the conditions are met. So, the sampling distribution model is $N(0.12,0.025)$ . Thus, we can apply the central limit theorem to describe the sampling distribution model for the sample proportion of children who are near-sighted.

(b)

To determine

To sketch and clearly label the sampling model based on the $68-95-99.7$ rule.

Explanation of Solution

It is given in the question that it is generally believed that near-sightedness affects about $12\%$ of all students. And a school district has registered $170$ incoming kindergarten children. Thus, the sketch of the sampling model based on the $68-95-99.7$ rule is as follows:

  

(c)

To determine

To explain how many of the incoming students might the school expect to be near-sighted.

Explanation of Solution

It is given in the question that it is generally believed that near-sightedness affects about $12\%$ of all students. And a school district has registered $170$ incoming kindergarten children. Thus, they might expect that the proportion of near-sighted students to be within two standard deviations of the mean. According to this model, this means that they should expect between $7\%\text{ and 17\%}$ of the student to be nearsighted.