Chapter 6 Confidence Intervals

Chapter Outline 6.1 Confidence Intervals for the Mean ( Known) 6.2 Confidence Intervals for the Mean ( Unknown) 6.3 Confidence Intervals for Population Proportions 6.4 Confidence Intervals for Variance and Standard Deviation .

Confidence Intervals for Variance and Standard Deviation Section 6.4 Confidence Intervals for Variance and Standard Deviation .

Section 6.4 Objectives How to interpret the chi-square distribution and use a chi-square distribution table How to construct and interpret confidence intervals for a population variance and standard deviation .

The Chi-Square Distribution The point estimate for 2 is s2 The point estimate for  is s s2 is the most unbiased estimate for 2 Estimate Population Parameter… with Sample Statistic Variance: σ2 s2 Standard deviation: σ s .

The Chi-Square Distribution 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 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. .

Properties of The Chi-Square Distribution Chi-square distribution is positively skewed. Chi-square distribution is different for each number of degrees of freedom. As degrees of freedom increase, the chi-square distribution approaches a normal distribution. .

Properties of The Chi-Square Distribution Chi-square distribution is positively skewed. Chi-square distribution is different for each number of degrees of freedom. As degrees of freedom increase, the chi-square distribution approaches a normal distribution. .

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. χ2 c The area between the left and right critical values is c. .

Example: Finding Critical Values for χ2 Find the critical values and for a 95% confidence interval when the sample size is 18. Solution: d.f. = n – 1 = 18 – 1 = 17 d.f. Each area in the table represents the region under the chi-square curve to the right of the critical value. Area to the right of χ2R = Area to the right of χ2L = .

Solution: Finding Critical Values for χ2 Table 6: χ2-Distribution 7.564 30.191 95% of the area under the curve lies between 7.564 and 30.191. .

Confidence Intervals for 2 and  Confidence Interval for 2: Confidence Interval for : The probability that the confidence intervals contain σ2 or σ is c, assuming that the estimation process is repeated a large number of times. .

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 .

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. .

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. .

Solution: Constructing a Confidence Interval Area to the right of χ2R = Area to the right of χ2L = The critical values are χ2R = 52.336 and χ2L = 13.121 .

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. .

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. .

Section 6.4 Summary Interpreted the chi-square distribution and used a chi-square distribution table Constructed and interpreted the chi-square distribution to construct a confidence interval for a population variance and standard deviation .