Question

A researcher surveyed 150 high school students and found that 68% played a musical insturment. What would be a reasonable range for the population mean?

Answer 1

The reasonable range for the population mean is (61%, 75%).

Explanation:

The interval estimate of a population parameter is an interval of values that consist of the values within which the true value of the parameter lies with a certain probability.

The mean of the sampling distribution of sample proportion is,
`\hat p`.

One of the best interval estimate of population proportion is the 95% confidence interval for proportion,

`CI=\hat p \pm z_(\alpha/2)\sqrt{(\hat p(1-\hat p))/(n)}`

Given:

n = 150

`\hat p` = 0.68

The critical value of z for 95% confidence level is:

`z_(\alpha/2)=z_(0.05/2)=z_(0.025)=1.96`

Compute the 95% confidence interval for proportion as follows:

`CI=\hat p \pm z_(\alpha/2)\sqrt{(\hat p(1-\hat p))/(n)}`

`=0.68\pm1.96\sqrt{(0.68(1-0.68))/(150)}\\\\=0.68\pm 0.0747\\\\=(0.6053, 0.7547)\\\\\approx (0.61, 0.75)`

Thus, the reasonable range for the population mean is (61%, 75%).