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

Section 6.4 Objectives Larson/Farber 4th ed 2  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

The Chi-Square Distribution Larson/Farber 4th ed 3  The point estimate for  2 is s 2  The point estimate for  is s  s 2 is the most unbiased estimate for  2 Estimate Population Parameter… with Sample Statistic Variance: σ 2 s2s2 Standard deviation: σ s

The Chi-Square Distribution Larson/Farber 4th ed 4  You can use the chi-square distribution to 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.

Properties of The Chi-Square Distribution Larson/Farber 4th ed 5 1. All chi-square values χ 2 are greater than or equal to zero. 2. 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 χ 2 -distribution with degrees of freedom equal to one less than the sample size. d.f. = n – 1 Degrees of freedom 3. The area under each curve of the chi-square distribution equals one.

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

Critical Values for χ 2 Larson/Farber 4th ed 7  There are two critical values for each level of confidence.  The value χ 2 R represents the right-tail critical value  The value χ 2 L represents the left-tail critical value. The area between the left and right critical values is c. χ2χ2 c

Example: Finding Critical Values for χ 2 Larson/Farber 4th ed 8 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. Area to the right of χ 2 R = Area to the right of χ 2 L = Each area in the table represents the region under the chi-square curve to the right of the critical value.

Solution: Finding Critical Values for χ 2 Larson/Farber 4th ed 9 Table 6: χ 2 -Distribution 90% of the area under the curve lies between and

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

Confidence Intervals for  2 and  Larson/Farber 4th ed 11 1.Verify that the population has a normal distribution. 2.Identify the sample statistic n and the degrees of freedom. 3.Find the point estimate s 2. 4.Find the critical value χ 2 R and χ 2 L that correspond to the given level of confidence c. Use Table 6 in Appendix B d.f. = n – 1 In WordsIn Symbols

Confidence Intervals for  2 and  Larson/Farber 4th ed 12 5.Find the left and right endpoints and form the confidence interval for the population variance. 6.Find the confidence interval for the population standard deviation by taking the square root of each endpoint. In WordsIn Symbols

Example: Constructing a Confidence Interval Larson/Farber 4th ed 13 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.

Solution: Constructing a Confidence Interval Larson/Farber 4th ed 14 The critical values are χ 2 R = and χ 2 L = Area to the right of χ 2 R = Area to the right of χ 2 L =

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

Solution: Constructing a Confidence Interval Larson/Farber 4th ed 16 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.

Section 6.4 Summary Larson/Farber 4th ed 17  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