Answer to Question #200783 in Statistics and Probability for Abe

Question

A sample of 60 Grade 9 students’ ages was obtained to estimate the mean age of all Grade 9 students. =15.3 years and the population variance is 16.

1.) What is the point estimate for ?

2.) Find the 95% confidence interval for .

3.) Find the 99% confidence interval for .

4.) What conclusions can you make based on each estimate?

Answer 1

1)

The sample mean is a point estimate of the population mean.

2)

$-1.96<\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}<1.96$

For sample mean:

$\mu-1.96\sigma/\sqrt{n}<\overline{x}<\mu+1.96\sigma/\sqrt{n}$

$15.3-1.96\cdot4/\sqrt{60}<\overline{x}<15.3+1.96\cdot4/\sqrt{60}$

$14.29<\overline{x}<16.31$

3)

For the 99% confidence interval:

$Z_{0.95}=\pm2.58$

$-2.58<\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}<2.58$

$15.3-2.58\cdot4/\sqrt{60}<\overline{x}<15.3+2.58\cdot4/\sqrt{60}$

$13.97<\overline{x}<16.63$

4)

Both intervals lie within one $\sigma$ about the mean of population, and the 99% confidence interval is larger than the 95% confidence interval for the sample mean.