1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 8.5 Estimating Population Variances

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.24: Finding Point Estimates for the Population Standard Deviation and Variance General Auto is testing the variance in the lengths of its windshield wiper blades. A sample of 12 windshield wiper blades is randomly selected, and the following lengths are measured in inches. 22.1 22.0 22.1 22.4 22.3 22.5 22.3 22.1 22.2 22.6 22.5 22.7 a.Find a point estimate for the population standard deviation. b.Find a point estimate for the population variance.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.24: Finding Point Estimates for the Population Standard Deviation and Variance (cont.) a.The sample standard deviation, s, is the most common point estimate for σ. According to the calculator, s ≈ 0.224958 ≈ 0.22. For review on how to calculate s, see Section 3.2. The point estimate for the population standard deviation is then 0.22 inches. b.The sample variance, s 2, is the best point estimate for σ 2. Using a calculator, we calculate s 2 ≈ (0.224958) 2 ≈ 0.05. The point estimate for the population variance is then 0.05.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Population Variances Confidence Interval for a Population Variance When the sample taken is a simple random sample and the population distribution is approximately normal, the confidence interval for a population variance is given by

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Population Variances Confidence Interval for a Population Variance (cont.) where n is the sample size, s 2 is the sample variance, and are the critical values for the level of confidence, c = 1 − , such that the area under the χ 2 -distribution with n  1 degrees of freedom to the right of is equal to and the area to the right of is equal to

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Population Variances Confidence Interval for a Population Standard Deviation When the sample taken is a simple random sample and the population distribution is approximately normal, the confidence interval for a population standard deviation is given by

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Confidence Interval for a Population Standard Deviation Confidence Interval for a Population Standard Deviation (cont.) where n is the sample size, s 2 is the sample variance, and are the critical values for the level of confidence, c = 1 − , such that the area under the χ 2 -distribution with n  1 degrees of freedom to the right of is equal to and the area to the right of is equal to

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Constructing a Confidence Interval for a Population Variance (or Standard Deviation) 1.Find the point estimate, s 2 (or s). 2.Based on the level of confidence given, calculate 3.Use the χ 2 -distribution table to find the critical values, for a distribution with n  1 degrees of freedom.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Constructing a Confidence Interval for a Population Variance (or Standard Deviation) Constructing a Confidence Interval for a Population Variance (or Standard Deviation) (cont.) 4.Substitute the necessary values into the formula for the confidence interval.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.25: Constructing a Confidence Interval for a Population Variance A commercial bakery is testing the variance in the weights of the cookies it produces. A random sample of 15 cookies is chosen; the weights of the cookies are measured in grams and found to have a variance of 3.4. Build a 95% confidence interval for the variance of the weights of all cookies produced by the company. Solution Step 1:Find the point estimate, s 2 (or s). We are given the sample variance in the problem: s 2 = 3.4.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.25: Constructing a Confidence Interval for a Population Variance (cont.) Step 2:Based on the level of confidence given, calculate Because c = 0.95, we know that  = 1 − 0.95 = 0.05. Then

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.25: Constructing a Confidence Interval for a Population Variance (cont.) Step 3:Find the critical values, Using the χ 2 -distribution table with df = n – 1 = 15 – 1 = 14 degrees of freedom, we see that and

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.25: Constructing a Confidence Interval for a Population Variance (cont.) Area to the Right of the Critical Value of χ 2 df0.9950.9900.9750.9500.9000.1000.0500.0250.0100.005 112.6033.0533.8164.5755.57817.27519.67521.92024.72526.757 123.0743.5714.4045.2266.30418.54921.02623.33726.21728.300 133.5654.1075.0095.8927.04219.81222.36224.73627.68829.819 144.0754.6605.6296.5717.79021.06423.68526.11929.14131.319 154.6015.2296.2627.2618.54722.30724.99627.48830.57832.801 165.1425.8126.9087.9629.31223.54226.29628.84532.00034.267

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.25: Constructing a Confidence Interval for a Population Variance (cont.) Step 4:Substitute the necessary values into the formula for the confidence interval. Substituting into the formula for a confidence interval for a population variance gives us the following.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.25: Constructing a Confidence Interval for a Population Variance (cont.) Using interval notation, the confidence interval can also be written as (1.8, 8.5). The bakery estimates with 95% confidence that the variance in the weights of their cookies is between 1.8 and 8.5.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.26: Constructing a Confidence Interval for a Population Standard Deviation Consider again the bakery in the previous example. Suppose that the bakery needs to estimate the standard deviation of the weights of their cookies as well. Construct a 95% confidence interval for the standard deviation of the weights of all cookies produced at the bakery. Solution Remember the relationship between standard deviation and variance.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.26: Constructing a Confidence Interval for a Population Standard Deviation (cont.) To find the 95% confidence interval for standard deviation, simply take the square roots of the expressions used to find the endpoints of the 95% confidence interval for the variance. This gives us the following.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.26: Constructing a Confidence Interval for a Population Standard Deviation (cont.) Using interval notation, the confidence interval can also be written as (1.3, 2.9). The bakery can be 95% confident that the standard deviation of the weights of all cookies produced is between 1.3 and 2.9 grams.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.27: Constructing a Confidence Interval for a Population Variance A seed company is researching the consistency in the output of a new hybrid tomato plant that was recently developed. The weights of a random sample of 75 tomatoes produced by the hybrid plant are measured in pounds, and the sample has a variance of 0.0225. Construct a 90% confidence interval for the variance in weights for all tomatoes produced by the new hybrid plant.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.27: Constructing a Confidence Interval for a Population Variance (cont.) Solution Step 1:Find the point estimate, s 2 (or s). We are given the sample variance, s 2 = 0.0225. Step 2:Based on the level of confidence given, calculate Because c = 0.90, we know that  = 1 − 0.90 = 0.10. Then

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.27: Constructing a Confidence Interval for a Population Variance (cont.) Step 3:Find the critical values, We need to find the critical values for the χ 2 -distribution with df = n – 1 = 75 – 1 = 74 degrees of freedom, but 74 is not one of the numbers of degrees of freedom listed in the table. Using the closest value available in the χ 2 -distribution table, df = 70, we see that

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.27: Constructing a Confidence Interval for a Population Variance (cont.) Area to the Right of the Critical Value of χ 2 df0.9950.9900.9750.9500.9000.1000.0500.0250.0100.005 4020.70722.16424.43326.50929.05151.80555.75859.34263.69166.766 5027.99129.70732.35734.76437.68963.16767.50571.42076.15479.490 6035.53437.48540.48243.18846.45974.39779.08283.29888.37991.952 7043.27545.44248.75851.73955.32985.52790.53195.023100.425104.215 8051.17253.54057.15360.39164.27896.578101.879106.629112.329116.321 9059.19661.75465.64769.12673.291107.565113.145118.136124.116128.299

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.27: Constructing a Confidence Interval for a Population Variance (cont.) Step 4:Substitute the necessary values into the formula for the confidence interval. Substituting into the formula for a confidence interval for a population variance gives us the following.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.27: Constructing a Confidence Interval for a Population Variance (cont.) Using interval notation, the confidence interval can also be written as (0.0184, 0.0322). Therefore, the seed company can estimate with 90% confidence that the true variance in weights of the new hybrid tomatoes is between 0.0184 and 0.0322.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation Let’s consider again the scenario of the seed company given in Example 8.27. A new random sample of 86 tomatoes is taken from the hybrid plants and found to have a standard deviation of 0.1400 pounds. Construct a 99% confidence interval to estimate the standard deviation in the weights of all tomatoes produced by the new hybrid plant. Solution Step 1:Find the point estimate, s 2 (or s). We are given the sample standard deviation, s = 0.1400.

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation (cont.) Step 2:Based on the level of confidence given, calculate Because c = 0.99, we know that  = 1 − 0.99 = 0.01. Then

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation (cont.) Step 3:Find the critical values, We need to find the critical values for the χ 2 -distribution with df = n – 1 = 86 – 1 = 85 degrees of freedom, but we see that not only is 85 degrees of freedom not listed in the table, but it is exactly halfway between the two closest values available, 80 and 90 degrees of freedom. Therefore for each critical value, we will need to find the mean of the values listed for 80 and 90 degrees of freedom in order to obtain the value we need.

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation (cont.) Area to the Right of the Critical Value of χ 2 df0.9950.9900.9750.9500.9000.1000.0500.0250.0100.005 4020.70722.16424.43326.50929.05151.80555.75859.34263.69166.766 5027.99129.70732.35734.76437.68963.16767.50571.42076.15479.49 6035.53437.48540.48243.18846.45974.39779.08283.29888.37991.952 7043.27545.44248.75851.73955.32985.52790.53195.023100.425104.215 8051.17253.54057.15360.39164.27896.578101.879106.629112.329116.321 9059.19661.75465.64769.12673.291107.565113.145118.136124.116128.299 10067.32870.06574.22277.92982.358118.498124.342129.561135.807140.169

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation (cont.) Taking the mean of 116.321 and 128.299, we obtain a critical value of for 85 degrees of freedom. Taking the mean of 51.172 and 59.196, we obtain a critical value of

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation (cont.) Step 4:Substitute the necessary values into the formula for the confidence interval. Substituting into the formula for a confidence interval for a population standard deviation gives us the following.

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation (cont.)

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.28: Constructing a Confidence Interval for a Population Standard Deviation (cont.) Using interval notation, the confidence interval can also be written as (0.1167, 0.1738). Therefore, the seed company can estimate with 99% confidence that the true standard deviation in weights of all tomatoes produced by the new hybrid plant is between 0.1167 and 0.1738 pounds.

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.29: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Standard Deviation A market researcher wants to estimate the standard deviation of home prices in a metropolitan area in the Northeast. She needs to be 99% confident that her sample standard deviation is within 5% of the true population standard deviation. Assuming that the home prices in that area are normally distributed, what is the minimum number of home prices she must acquire?

34. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.29: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Standard Deviation (cont.) Solution According to the table, we see that, in order to be 99% confident that the sample standard deviation will be within 5% of the true population standard deviation, the minimum sample size required is 1337.

35. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.29: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Standard Deviation (cont.) Minimum Sample Sizes for Estimating Standard Deviation s Is within This Percentage of the Value of  Minimum Sample Size Needed for 95% Level of Confidence Minimum Sample Size Needed for 99% Level of Confidence 1%19,20633,220 5%7691337 10%193337 20%4986 30%2239 40%1323 50%915

36. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.29: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Standard Deviation (cont.) Therefore, the market researcher must include at least 1337 home prices in order for her study to have the level of precision that she needs.