Question

It is believed that nearsightedness affects about 12​% of all children. A kindergarten has registered 198 incoming children. Complete parts​ a) through​ c). ​a) Can the central limit theorem be applied to describe the sampling distribution for the sample proportion of children who are​ nearsighted? Check the conditions and discuss any assumptions you need to make.

Answer 1

As the sample size is large enough, i.e. n = 198 > 30 the central limit theorem can be applied to describe the sampling distribution for the sample proportion of children who are​ nearsighted.

Explanation:

Let the random variable p denote the proportion of children affected by nearsightedness.

The previously known proportion was, p = 0.12.

A random sample of n = 198 children are selected.

According to the Central limit theorem, if from an unknown population large samples of sizes n > 30, are selected and the sample proportion for each sample is computed then the sampling distribution of sample proportion follows a Normal distribution.

The mean of this sampling distribution of sample proportion is:

`\mu_(\hat p)=p`

The standard deviation of this sampling distribution of sample proportion is:

`\sigma_(\hat p)=\sqrt{(p(1-p))/(n)}`

As the sample size is large enough, i.e. n = 198 > 30 the central limit theorem can be applied to describe the sampling distribution for the sample proportion of children who are​ nearsighted.