Questions: SAT Scores Estimate the variance in mean mathematics SAT scores by state, using the randomly selected scores listed below. Estimate with 95% confidence. Assume the variable is normally distributed. Round the sample variance and final answers to one decimal place. 211, 565, 543, 209, 572, 502, 550, 500, 469, 499, 515 Send data to Excel <σ²<

SAT Scores Estimate the variance in mean mathematics SAT scores by state, using the randomly selected scores listed below. Estimate with 95% confidence. Assume the variable is normally distributed. Round the sample variance and final answers to one decimal place. 211, 565, 543, 209, 572, 502, 550, 500, 469, 499, 515 Send data to Excel <σ²<
Transcript text: SAT Scores Estimate the variance in mean mathematics SAT scores by state, using the randomly selected scores listed below. Estimate with $95 \%$ confidence. Assume the variable is normally distributed. Round the sample variance and final answers to one decimal place. \begin{tabular}{llllll} 211 & 565 & 543 & 209 & 572 & 502 \\ 550 & 500 & 469 & 499 & 515 & \end{tabular} Send data to Excel $\square$ $<\sigma^{2}<$ $\square$
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Solution

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Solution Steps

Step 1: Calculate the Mean

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

\[ \mu = \frac{\sum x_i}{n} = \frac{5135}{11} = 466.8 \]

Step 2: Calculate the Sample Variance

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

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 17104.0 \]

Step 3: Calculate the Standard Deviation

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

\[ \sigma = \sqrt{17104.0} = 130.8 \]

Step 4: Calculate the 95% Confidence Interval for Variance

Using the chi-square distribution, the 95% confidence interval for the variance is determined to be:

\[ 87266.9 < \sigma^2 < -87266.9 \]

Final Answer

The sample variance and the 95% confidence interval for the variance of the mean mathematics SAT scores by state are:

\[ \text{Sample Variance: } 17104.0 \] \[ \text{95% Confidence Interval: } 87266.9 < \sigma^2 < -87266.9 \]

Thus, the final boxed answer is:

\[ \boxed{87266.9 < \sigma^2 < -87266.9} \]

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