Question

4. The mean age of 50 randomly selected teachers in southern California was 42.5. If it is known from other studies that the standard deviation of ages of all teachers is 8.75, (a) (2 points) Find the critical value a/2 or ta/2 that properly applies in this problem for 95% confidence interval. 05 (b) (2 points) Find the 95% confidence interval for the mean age of all teachers in southern California. Page (b) (e) (2 points) Find the margin of error for this confidence interval. (c) Total Points: 50 Page 2 of 4 Study Guide 25

Answer 1

(a) Since population standard deviation is known , the appropriate critical value is :

                                

22 2 1-0.95 = 20.025 1.96

(b) The formula for the 95% confidence interval for mean is :

    $[\bar{x}-z_{\frac{1-0.95}{2}}*\frac{\sigma}{\sqrt{n}},\bar{x}+z_{\frac{1-0.95}{2}}*\frac{\sigma}{\sqrt{n}}]\\= [42.5-1.96*\frac{8.75}{\sqrt{50}},42.5+1.96*\frac{8.75}{\sqrt{50}}]$

$=[40.0746,44.9254]$

(c) Margin of error :-      

              $z_{\frac{1-0.95}{2}}*\frac{\sigma}{\sqrt{n}}\\\\= 1.96*\frac{8.75}{\sqrt{50}}\\=2.4254$