1. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 1 Slides by John Loucks St. Edward’s University

2. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 2 2 Chapter 8 Interval Estimation n n Population Mean:  Known n n Population Mean:  Unknown n n Population Proportion

3. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 3 A point estimator cannot be expected to provide the exact value of the population parameter. An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. Point Estimate +/- Margin of Error The purpose of an interval estimate is to provide information about how close the point estimate is to the value of the parameter. Margin of Error and the Interval Estimate

4. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 4 The general form of an interval estimate of a population mean is Margin of Error and the Interval Estimate

5. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 5 Interval Estimate of a Population Mean:  Known n n In order to develop an interval estimate of a population mean, the margin of error must be computed using either: the population standard deviation , or the sample standard deviation s n n  is rarely known exactly, but often a good estimate can be obtained based on historical data or other information. n n We refer to such cases as the  known case.

6. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 6 6   /2 Interval Estimate of a Population Mean:  Known

7. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 7   /2 interval does not include  interval includes  Interval Estimate of a Population Mean:  Known

8. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 8 8 Interval Estimate of a Population Mean:  Known n Interval Estimate of 

9. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 9 9 Interval Estimate of a Population Mean:  Known n n Values of z a/2 for the Most Commonly Used Confidence Levels 90%.10.05.9500 1.645 Confidence Table Level   /2 Look-up Area z  /2 95%.05.025.9750 1.960 99%.01.005.9950 2.576

10. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 10 Meaning of Confidence We say that this interval has been established at the 90% confidence level. The value.90 is referred to as the confidence coefficient.

11. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 11 Interval Estimate of a Population Mean:  Known n n Example: Discount Sounds Discount Sounds has 260 retail outlets throughout the United States. The firm is evaluating a potential location for a new outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. A sample of size n = 36 was taken; the sample mean income is $41,100. The population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500, and the confidence coefficient to be used in the interval estimate is.95.

12. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 12 The margin of error is: Thus, at 95% confidence, the margin of error is $1,470. n n Example: Discount Sounds Interval Estimate of a Population Mean:  Known

13. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 13 Interval Estimate of a Population Mean:  Known Interval estimate of  is: We are 95% confident that the interval contains the population mean. $41,100 + $1,470 or $39,630 to $42,570 n n Example: Discount Sounds

14. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 14 Interval Estimate of a Population Mean:  Known 90% 1,234 39,866 to 42,334 Confidence Margin Level of Error Interval Estimate 95% 1,470 39,630 to 42,570 99% 1,932 39,168 to 43,032 n n Example: Discount Sounds In order to have a higher degree of confidence, the margin of error and thus the width of the confidence interval must be larger.

15. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 15 Using Excel to Construct a Confidence Interval:  Known Case Confidence interval in this case can be most easily done by Excel. See Excel Demonstration posted in the chapter module in D2L. Of course you should understand the manual procedure to do it with Excel.

16. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 16 n n Adequate Sample Size In most applications, a sample size of n = 30 is adequate. If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended. Interval Estimate of a Population Mean:  Known

17. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 17 n n Adequate Sample Size (continued) If the population is believed to be at least approximately normal, a sample size of less than 15 can be used. If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice. Interval Estimate of a Population Mean:  Known

18. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 18 Interval Estimate of a Population Mean:  Unknown n n If an estimate of the population standard deviation  cannot be developed prior to sampling, we use the sample standard deviation s to estimate . n n This is the  unknown case. n n In this case, the interval estimate for  is based on the t distribution. n n (We’ll assume for now that the population is normally distributed.)

19. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 19 William Gosset, writing under the name “Student”, is the founder of the t distribution. t Distribution Gosset was an Oxford graduate in mathematics and worked for the Guinness Brewery in Dublin. He developed the t distribution while working on small-scale materials and temperature experiments.

20. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 20 The t distribution is a family of similar probability distributions. t Distribution A specific t distribution depends on a parameter known as the degrees of freedom. Degrees of freedom refer to the number of independent pieces of information that go into the computation of s.

21. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 21 t Distribution A t distribution with more degrees of freedom has less dispersion. As the degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller.

22. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 22 t Distribution Standard normal distribution Standard normal distribution t distribution (20 degrees of freedom) t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) t distribution (10 degrees of freedom) 0 z, t

23. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 23 For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value. t Distribution

24. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 24 DegreesArea in Upper Tail of Freedom.20.10.05.025.01.005....... 50.8491.2991.6762.0092.4032.678 60.8481.2961.6712.0002.3902.660 80.8461.2921.6641.9902.3742.639 100.8451.2901.6601.9842.3642.626.8421.2821.6451.9602.3262.576 t Distribution Standard normal z values Standard normal z values

25. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 25 n Interval Estimate Interval Estimate of a Population Mean:  Unknown

26. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 26 A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 16 one-bedroom apartments within a half-mile of campus resulted in a sample mean of $750 per month and a sample standard deviation of $55. n n Example: Apartment Rents Let us provide a 95% confidence interval estimate of the mean rent per month for the population of one- bedroom apartments within a half-mile of campus. We will assume this population to be normally distributed. Interval Estimate of a Population Mean:  Unknown

27. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 27 DegreesArea in Upper Tail of Freedom.20.100.050.025.010.005 15.8661.3411.7532.1312.6022.947 16.8651.3371.7462.1202.5832.921 17.8631.3331.7402.1102.5672.898 18.8621.3301.7342.1012.5202.878 19.8611.3281.7292.0932.5392.861....... At 95% confidence,  =.05, and  /2 =.025. In the t distribution table we see that t.025 = 2.131. t.025 is based on n - 1 = 16 - 1 = 15 degrees of freedom. Interval Estimate of a Population Mean:  Unknown

28. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 28 We are 95% confident that the mean rent per month for the population of one-bedroom apartments within a half-mile of campus is between $720.70 and $779.30. n n Interval Estimate Margin of Error Margin of Error Interval Estimate of a Population Mean:  Unknown

29. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 29 Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Descriptive Statistics from the list of Analysis Tools Step 4 When the Descriptive Statistics dialog box appears: (see details on next slide) Using Excel’s Descriptive Statistics Tool

30. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 30 Using Excel’s Descriptive Statistics Tool Confidence interval in this case can be most easily done by Excel. See Excel Demonstration posted in the chapter module in D2L. Of course you should understand the manual procedure to do it with Excel

31. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 31 Interval Estimate of a Population Mean:  Unknown n n Adequate Sample Size If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended.

32. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 32 n n Adequate Sample Size (continued) If the population is believed to be at least approximately normal, a sample size of less than 15 can be used. If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice. Interval Estimate of a Population Mean:  Unknown

33. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 33 Summary of Interval Estimation Procedures for a Population Mean Can the population standard deviation  be assumed known ? YesNo  Known Case  Unknown Case Use the sample standard deviation s to estimate  Use

34. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 34 Let E = the desired margin of error. E is the amount added to and subtracted from the point estimate to obtain an interval estimate. Sample Size for an Interval Estimate of a Population Mean If a desired margin of error is selected prior to sampling, the sample size necessary to satisfy the margin of error can be determined.

35. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 35 The general form of an interval estimate of a population proportion is Interval Estimate of a Population Proportion

36. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 36 Interval Estimate of a Population Proportion n Interval Estimate

37. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 37 Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office seekers informed of their position in a race. Using telephone surveys, PSI interviewers ask registered voters who they would vote for if the election were held that day. n n Example: Political Science, Inc. Interval Estimate of a Population Proportion

38. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 38 In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor a particular candidate. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favor the candidate. n n Example: Political Science, Inc. Interval Estimate of a Population Proportion

39. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 39 End of Chapter 8

40. © 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 40 Using Excel for Interval Estimate of a Population Proportion Confidence interval in this case can be most easily done by Excel. See Excel Demonstration posted in the chapter module in D2L. Of course you should understand the manual procedure to do it with Excel