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Section 6.4

Section 6.4. Confidence Intervals for Variance and Standard Deviation. Section 6.4 Objectives. Interpret the chi-square distribution and use a chi-square distribution table Use the chi-square distribution to construct a confidence interval for the variance and standard deviation.

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Section 6.4

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  1. Larson/Farber 4th ed Section 6.4 Confidence Intervals for Variance and Standard Deviation

  2. Section 6.4 Objectives • Interpret the chi-square distribution and use a chi-square distribution table • Use the chi-square distribution to construct a confidence interval for the variance and standard deviation Larson/Farber 4th ed

  3. The Chi-Square Distribution • The pointestimate for 2is s2 • The point estimate for is s • s2 is the most unbiased estimate for 2 Larson/Farber 4th ed

  4. The Chi-Square Distribution • You can use the chi-square distributionto construct a confidence interval for the variance and standard deviation. • If the random variable x has a normal distribution, then the distribution of forms a chi-square distribution for samples of any size n > 1. Larson/Farber 4th ed

  5. Properties of The Chi-Square Distribution • All chi-square values χ2 are greater than or equal to zero. • The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for 2, use the χ2-distribution with degrees of freedom equal to one less than the sample size. • d.f. = n – 1 Degrees of freedom • The area under each curve of the chi-square distribution equals one. Larson/Farber 4th ed

  6. Properties of The Chi-Square Distribution • Chi-square distributions are positively skewed. chi-square distributions Larson/Farber 4th ed

  7. Critical Values for χ2 • There are two critical values for each level of confidence. • The value χ2R represents the right-tail critical value • The value χ2L represents the left-tail critical value. c The area between the left and right critical values is c. χ2 Larson/Farber 4th ed

  8. Area to the right of χ2R = • Area to the right of χ2L = Example: Finding Critical Values for χ2 Find the critical values and for a 90% confidence interval when the sample size is 20. • Solution: • d.f. = n – 1 = 20 – 1 = 19 d.f. • Each area in the table represents the region under the chi-square curve to the right of the critical value. Larson/Farber 4th ed

  9. Solution: Finding Critical Values for χ2 Table 6: χ2-Distribution 30.144 10.117 90% of the area under the curve lies between 10.117 and 30.144 Larson/Farber 4th ed

  10. Confidence Intervals for 2 and  • Confidence Interval for 2: • Confidence Interval for : • The probability that the confidence intervals contain σ2 orσ is c. Larson/Farber 4th ed

  11. Confidence Intervals for 2 and  In Words In Symbols • Verify that the population has a normal distribution. • Identify the sample statistic n and the degrees of freedom. • Find the point estimate s2. • Find the critical value χ2R and χ2L that correspond to the given level of confidence c. d.f. = n – 1 Use Table 6 in Appendix B Larson/Farber 4th ed

  12. Confidence Intervals for 2 and  In Words In Symbols • Find the left and right endpoints and form the confidence interval for the population variance. • Find the confidence interval for the population standard deviation by taking the square root of each endpoint. Larson/Farber 4th ed

  13. Example: Constructing a Confidence Interval You randomly select and weigh 30 samples of an allergy medicine. The sample standard deviation is 1.20 milligrams. Assuming the weights are normally distributed, construct 99% confidence intervals for the population variance and standard deviation. • Solution: • d.f. = n – 1 = 30 – 1 = 29 d.f. Larson/Farber 4th ed

  14. Area to the right of χ2R = • Area to the right of χ2L = Solution: Constructing a Confidence Interval • The critical values areχ2R = 52.336 and χ2L = 13.121 Larson/Farber 4th ed

  15. Solution: Constructing a Confidence Interval Confidence Interval for 2: Left endpoint: Right endpoint: 0.80 < σ2 < 3.18 With 99% confidence you can say that the population variance is between 0.80 and 3.18 milligrams. Larson/Farber 4th ed

  16. Solution: Constructing a Confidence Interval Confidence Interval for : 0.89 < σ < 1.78 With 99% confidence you can say that the population standard deviation is between 0.89 and1.78 milligrams. Larson/Farber 4th ed

  17. Section 6.4 Summary • Interpreted the chi-square distribution and used a chi-square distribution table • Used the chi-square distribution to construct a confidence interval for the variance and standard deviation Larson/Farber 4th ed

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