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Chapter 7: Problem 14

SAT Scores 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. $$ \begin{array}{llllll}{490} & {502} & {211} & {209} & {499} & {565} \\ {469} & {543} & {572} & {550} & {515} & {500}\end{array} $$

Short Answer

Expert verified
The variance is estimated to be between 605.13 and 5301.92 with 99% confidence.

Step by step solution

01

Calculate the Mean

Find the mean of the given SAT scores. Add all the scores together and divide by the number of scores. The scores are: 490, 502, 211, 209, 499, 565, 469, 543, 572, 550, 515, and 500. There are 12 scores in total. The sum of these scores is 5625. Therefore, the mean is \( \bar{x} = \frac{5625}{12} \approx 468.75 \).
02

Calculate the Variance

To calculate the variance, subtract the mean from each score, square the result, and then find the average of these squared differences. Use the formula \( s^2 = \frac{\Sigma (x_i - \bar{x})^2}{n-1} \). The squared differences are calculated as follows: \((490 - 468.75)^2, (502 - 468.75)^2, \ldots, (500 - 468.75)^2\). Sum these squared differences and you'll find approximately \( s^2 = 14713.08 \).
03

Identify Chi-Square Values for Confidence Interval

Using a Chi-Square distribution table, identify the critical chi-square values for a 99% confidence level with \( n-1 \) degrees of freedom. For 11 degrees of freedom, find \( \chi^2_{\alpha/2} \) and \( \chi^2_{1-\alpha/2} \). At 99% confidence, these values are approximately \( \chi^2_{0.005} = 3.053 \) and \( \chi^2_{0.995} = 26.757 \).
04

Compute Confidence Interval for Variance

Using the calculated variance and chi-square values, compute the confidence interval for the variance: \[ \left( \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \right) \]. Here, \( n = 12 \), \( s^2 = 1471.08 \), and \( n-1 = 11 \). This results in \( \left( \frac{11 \times 1471.08}{26.757}, \frac{11 \times 1471.08}{3.053} \right) \approx (605.13, 5301.92) \).
05

Conclusion

The variance in SAT scores for the sample is estimated with 99% confidence to lie between 605.13 and 5301.92. This means we are 99% confident that the true population variance falls within this range based on the given sample data.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

SAT Scores
SAT scores are standardized test scores used widely for college admissions in the United States. They aim to measure a student's readiness for college and offer a common data point to compare applicants. The SAT comprises mostly of mathematics, reading, and writing sections, with scores ranging between 400 and 1600. To estimate the variance over different states, sampled scores from these tests can be used. These scores capture the diversity and represent the ability distribution among students across regions.
  • SAT scores offer a snapshot of educational achievement in maths, reading, etc.
  • They help in understanding the test preparedness across different U.S. states.
  • Variance analysis on these scores helps infer differences in educational outcomes.
Confidence Interval
A confidence interval provides a range of values within which we can say, with a certain level of confidence, that a parameter of interest lies. For example, in estimating variance, a 99% confidence interval suggests that 99% of such computed intervals will contain the true variance. Constructing a confidence interval requires understanding of sampling distribution and variability. First, identify the level of confidence, such as 99%, which determines the critical values from a distribution (such as Chi-Square).
  • Confidence intervals give a range for population parameters based on sample data.
  • The width and position of confidence intervals convey information about estimate precision.
  • 99% confidence means the interval will capture the true value in nearly all samples.
Variance
Variance is a measure that tells us how much data points differ from the mean. It's a crucial concept in statistics used for various analyses, including finding confidence intervals. The variance is calculated as the average of the squared differences from the mean. This helps us understand the spread and variability of data points. In the context of SAT scores, variance helps us to see how spread out the students' scores are around the average.
  • Variance involves squaring deviations from the mean to avoid negative differences canceling out.
  • High variance indicates scores are widely spread; low variance suggests they are clustered.
  • Understanding variance in SAT scores helps educators address broad performance trends.
Chi-Square Distribution
The Chi-Square distribution is often used in statistics for hypothesis testing and confidence interval estimation. It is particularly useful when dealing with variances of normally distributed populations. In the estimation of variance, knowing the degrees of freedom鈥攂ased on the sample size鈥攊s essential to find critical values from the Chi-Square distribution table. These critical values are then used to construct the confidence interval.
  • Chi-Square distribution is skewed to the right, non-negative, and depends on the degrees of freedom.
  • This distribution becomes symmetrical with a large sample size (higher degrees of freedom).
  • It helps in constructing confidence intervals for variance by providing critical values.
  • Critical values from Chi-Square tables help determine the bounds of confidence intervals.

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