Questions: Estimate the variance in mean mathematics SAT scores by state, using the randomly selected scores listed below. Estimate with 99% confidence. Assume the variable is normally distributed. Round the sample variance and final answers to one decimal place. 490 209 543 515 499 500 572 565 550 469 211

Solution Steps

The mean \( \mu \) of the given SAT scores is calculated as follows:

\[ \mu = \frac{\sum x_i}{n} = \frac{5123}{11} \approx 465.7 \]

The sample variance \( \sigma^2 \) is computed using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 17032.6 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{17032.6} \approx 130.5 \]

The confidence interval for the variance of a single population with unknown population mean is given by:

\[ \left(\frac{(n - 1)s^2}{\chi^2_{\alpha/2}}, \frac{(n - 1)s^2}{\chi^2_{1 - \alpha/2}}\right) \]

Substituting the values:

\[ CI = \left(\frac{(11 - 1) \times 17032.6}{\chi^2_{\alpha/2}}, \frac{(11 - 1) \times 17032.6}{\chi^2_{1 - \alpha/2}}\right) \]

This results in:

\[ CI \approx (6762.1, 79006.2) \]

Final Answer