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

Answer:

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

Step-by-step 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,

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

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:

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

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

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