Answer to Question #318471 in Statistics and Probability for Italia

Question

1.   A sample of 60 grade 11 students’ ages was obtained to estimate the mean age of all grade 11 students.  years and the population variance is 16.

a.   What is the point estimate for μ?

b.   Find the 95% confidence interval for μ.

c.    Find the 99% confidence interval for μ.

d.   What conclusions can you make based on each estimate.

Answer 1

$a:\hat{\mu}=\bar{x}=11\\b:\left( \bar{x}-\frac{\sigma}{\sqrt{n}}z_{\frac{1+\gamma}{2}},\bar{x}+\frac{\sigma}{\sqrt{n}}z_{\frac{1+\gamma}{2}} \right) =\left( 11-\frac{4}{\sqrt{60}}1.960,11+\frac{4}{\sqrt{60}}1.960 \right) =\\=\left( 9.98786,12.0121 \right) \\c: \left( \bar{x}-\frac{\sigma}{\sqrt{n}}z_{\frac{1+\gamma}{2}},\bar{x}+\frac{\sigma}{\sqrt{n}}z_{\frac{1+\gamma}{2}} \right) =\left( 11-\frac{4}{\sqrt{60}}\cdot 2.576,11+\frac{4}{\sqrt{60}}\cdot 2.576 \right) =\\=\left( 9.66976,12.3302 \right) \\d:Most\,\,students\,\,are\,\,between\,\,9 and\,\,12 years, 95\% are\,\,between\,\,9.99 and\,\,12.01,\\99\% are\,\,between\,\,9.67 and\,\,12.33$