1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 8.1 Estimating Population Means (  Known) And some added content by D.R.S., University of Cordele

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Population Means A point estimate is a single-number estimate of a population parameter. An unbiased estimator is a point estimate that does not consistently underestimate or overestimate the population parameter.

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. HAWKES LEARNING SYSTEMS math courseware specialists Confidence Interval for Population Means: Confidence Intervals 8.1 Introduction to Estimating Population Means E is the Margin of Error We claim that μ is between these values. We claim that μ is in this (low,high) interval

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.1: Finding a Point Estimate for a Population Mean Find the best point estimate for the population mean of test scores on a standardized biology final exam. The following is a simple random sample taken from the population of test scores. 45 68 72 91 100 71 69 83 86 55 89 97 76 68 92 75 84 70 81 90 85 74 88 99 76 91 93 85 96 100

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.1: Finding a Point Estimate for a Population Mean (cont.) Solution The best point estimate for the population mean is a sample mean because it is an unbiased estimator. The sample mean for the given sample of test scores is Thus, the best point estimate for the population mean of test scores on this standardized exam is 81.6.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Population Means An interval estimate is a range of possible values for a population parameter. The level of confidence is the probability that the interval estimate contains the population parameter. A confidence interval is an interval estimate associated with a certain level of confidence.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Estimating Population Means The margin of error, or maximum error of estimate, E, is the largest possible distance from the point estimate that a confidence interval will cover.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.2: Constructing a Confidence Interval with a Given Margin of Error A college student researching study habits collects data from a random sample of 250 college students on her campus and calculates that the sample mean is hours per week. If the margin of error for her data using a 95% level of confidence is E = 0.6 hours, construct a 95% confidence interval for her data. Interpret your results.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.2: Constructing a Confidence Interval with a Given Margin of Error (cont.) Solution The best point estimate for the population mean is a sample mean, so use as the point estimate for this population parameter. We are given the value of the margin of error for our confidence interval, E = 0.6. As indicated in Figure 8.1, the endpoints of a confidence interval are found by subtracting E from and adding E to the point estimate. To find the lower endpoint, subtract the margin of error from the sample mean; that is, calculate

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.2: Constructing a Confidence Interval with a Given Margin of Error (cont.) Thus, the lower endpoint is calculated as follows. To find the upper endpoint, add the margin of error to the sample mean; that is, calculate Thus, the upper endpoint is calculated as follows.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.2: Constructing a Confidence Interval with a Given Margin of Error (cont.) Therefore, the confidence interval ranges from 15.1 to 16.3. The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.2: Constructing a Confidence Interval with a Given Margin of Error (cont.) The interpretation of our confidence interval is that we are 95% confident that the true population mean for the number of hours per week that students on this campus spend studying is between 15.1 and 16.3 hours. But this was too easy. They handed us the value of E. We did minus on one side of the mean. We did plus on the other side of the mean. We really want to know “Where does that E value come from?” As the old saying goes, “Give a man the value of E and he will calculate one confidence interval; teach a man how to find E and he will enjoy statistics for a lifetime.”

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Using the Standard Normal Distribution to Estimate a Population Mean Margin of Error of a Confidence Interval for a Population Mean (  Known) When the population standard deviation is known, the sample taken is a simple random sample, and either the sample size is at least 30 or the population distribution is approximately normal, the margin of error of a confidence interval for a population mean is given by

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Using the Standard Normal Distribution to Estimate a Population Mean Margin of Error of a Confidence Interval for a Population Mean (  Known) (cont.) Where is the critical value for the level of confidence, c = 1 − , such that the area under the standard normal distribution to the right of is equal to  is the population standard deviation, and n is the sample size.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. A convenient table for common critical values Level of Confidencez α/2 0.80 (or 80%)1.28 0.85 (or 85%)1.44 0.90 (or 90%)1.645 0.95 (or 95%)1.96 0.98 (or 98%)2.33 0.99 (or 99%)2.575 You already know how to get these values. For example: 1.For an 80% level of confidence, 2.α = 1 – 80% = 1 – 0.8000 = 0.2000 3.α / 2 = 0.1000 4.What z (and –z) value causes area 0.8000 in the middle and 0.1000 in each tail? You already know how to get these values. For example: 1.For an 80% level of confidence, 2.α = 1 – 80% = 1 – 0.8000 = 0.2000 3.α / 2 = 0.1000 4.What z (and –z) value causes area 0.8000 in the middle and 0.1000 in each tail?

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.3: Finding the Margin of Error of a Confidence Interval for a Population Mean (  Known) Researchers want to estimate the mean monthly electricity bill in a large urban area using a simple random sample of 100 households. Assume that the population standard deviation is known to be $15.50. Find the margin of error for a 99% confidence interval. Round your answer to two decimal places.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.3: Finding the Margin of Error of a Confidence Interval for a Population Mean (  Known) (cont.) Solution First, refer to the table of critical z-values to find the critical value for c = 0.99. In the table, we see that for c = 0.99, the critical value is The population standard deviation is  = 15.50, and the sample size is n = 100.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.3: Finding the Margin of Error of a Confidence Interval for a Population Mean (  Known) (cont.) Substituting these values into the formula for the margin of error, we get the following. So the margin of error for this 99% confidence interval is approximately $3.99.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) In order to estimate the number of calls to expect at a new suicide hotline, volunteers contact a random sample of 35 similar hotlines across the nation and find that the sample mean is 42.0 calls per month. Construct a 95% confidence interval for the mean number of calls per month. Assume that the population standard deviation is known to be 6.5 calls per month.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) (cont.) Step 1: Find the point estimate. To find a confidence interval, you need to know a point estimate and a margin of error. The point estimate for a population mean is the sample mean. In this example, the sample mean is given to us as 42.0 calls per month. The margin of error must be calculated using the formula for population means with  known.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) (cont.) Step 2: Find the margin of error. To calculate the margin of error, refer to the table of critical z-values to find the critical value for c = 0.95. Note that is the critical value. The population standard deviation is  = 6.5, and the sample size is n = 35.

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) (cont.) Substituting these values into the formula for the margin of error, we get the following.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) (cont.) Step 3: Subtract the margin of error from and add the margin of error to the point estimate. To find the lower endpoint, subtract the margin of error from the sample mean. Thus, the lower endpoint is calculated as follows.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) (cont.) To find the upper endpoint, add the margin of error to the sample mean. Thus, the upper endpoint is calculated as follows.

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) (cont.) Therefore, the 95% confidence interval ranges from 39.8 to 44.2 calls per month. The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.4: Constructing a Confidence Interval for a Population Mean (  Known) (cont.) The interpretation of our confidence interval is that we are 95% confident that the true population mean for the numbers of calls to suicide hotlines across the nation is between 39.8 and 44.2 calls per month.

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.5: Constructing a Confidence Interval for a Population Mean (  Known) A toy company wants to know the mean number of new toys per child bought each year. Marketing strategists at the toy company collect data from the parents of 1842 randomly selected children. The sample mean is found to be 4.7 toys per child. Construct a 90% confidence interval for the mean number of new toys per child purchased each year. Assume that the population standard deviation is known to be 1.9 toys per child per year.

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.5: Constructing a Confidence Interval for a Population Mean (  Known) Solution Step 1: Find the point estimate. The point estimate for the population mean is the sample mean, which we are told is 4.7 toys per child. Step 2: Find the margin of error. Next, we need to calculate the margin of error. Since we want a 90% confidence interval, we can use the table to find the critical z-value, which is

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.5: Constructing a Confidence Interval for a Population Mean (  Known) Substituting the values we have into the formula for margin of error, we get the following.

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.5: Constructing a Confidence Interval for a Population Mean (  Known) Step 3: Subtract the margin of error from and add the margin of error to the point estimate. To find the lower endpoint, subtract the margin of error from the best point estimate, that is, the sample mean.

32. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.5: Constructing a Confidence Interval for a Population Mean (  Known) To find the upper endpoint, add the margin of error to the point estimate. Thus, the 90% confidence interval ranges from 4.6 to 4.8 new toys per child per year.

33. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.5: Constructing a Confidence Interval for a Population Mean (  Known) The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below. Therefore, we are 90% confident that the true population mean for the number of new toys per child bought each year is between 4.6 and 4.8 toys per child.

34. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.6: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (  Known) The owners of a local company that produces hand- knitted socks want to know, for women in their area, the average length of a woman’s foot, from toe to heel. They collect data from 431 randomly selected women. The sample mean is found to be 8.72 inches. Construct a 95% confidence interval for the mean length of a woman’s foot in the company's area. Assume the owners know that the population standard deviation is 0.36 inches.

35. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.6: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (  Known) (cont.) Solution Using a TI-83/84 Plus calculator, press, scroll to TESTS, and then select option 7:ZInterval, since the population standard deviation is known. We are given the sample statistics, so we highlight the Stats option, and enter the values for Ç, Ë, and n. For this example,  = 0.36, and n = 431. C-Level is the confidence level, which should be entered as a decimal. The confidence level for this example is 95%, so we enter 0.95.

36. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.6: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (  Known) (cont.) Highlight Calculate and press. The calculator screen then gives the confidence interval in interval form and also reiterates the sample mean and sample size.

37. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.6: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (  Known) (cont.) Thus, the 95% confidence interval ranges from 8.69 to 8.75 inches. The confidence interval can be written mathematically using either inequality symbols or interval notation, as shown below. Therefore, we are 95% confident that the true population mean for the length of a woman’s foot is between 8.69 and 8.75 inches.

38. Example 8.6: Using a TI-83/84 Plus Calculator to Find a Confidence Interval for a Population Mean (  Known) With Excel =CONFIDENCE.NORM( α, σ, n) α = level of significance = 1 – level of confidence σ = population standard deviation n = sample size

39. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Further words about ZInterval If you’re asked for a confidence interval, Use ZInterval for a normal distrib. situation. It’s easier than using the primitive formula The calculator keeps more decimal precision If the problem asks for a critical value of z, too, Then you have to use invNorm( or the printed table to answer that question. Make the right choice between Stats, if you’re given the mean, etc. Data, if you’re given a list of raw data

40. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Minimum Sample Size for Estimating a Population Mean The minimum sample size required for estimating a population mean at a given level of confidence with a particular margin of error is given by

41. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Minimum Sample Size for Estimating a Population Mean Minimum Sample Size for Estimating a Population Mean (cont.) Where is the critical value for the level of confidence, c = 1 − , such that the area under the standard normal distribution to the right of is equal to  is the population standard deviation, and E is the desired maximum margin of error.

42. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.8: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Mean Determine the minimum sample size needed if we wish to be 90% confident that the sample mean is within two units of the population mean. An estimate for the population standard deviation of 8.4 is available from a previous study. Solution From the information that we are given, we know the following values: c = 0.90 and  ≈ 8.4. The phrase “within two units” indicates that the desired maximum margin of error is two, so E = 2.

43. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.8: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Mean (cont.) Since we desire a 90% level of confidence, we can use the table of critical z-values to determine that Substituting these values into our formula for minimum sample size, we obtain the following.

44. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.8 with TI-84 Since we desire a 90% level of confidence, we can use the table of critical z-values to determine that TI-84 in one line: (1.645*8.4/2) 2 (used X 2 key) Or even more slick: : (-invNorm(0.05)*8.4/2) 2

45. Example 8.8 with Excel Line 1: typing the formula with all numbers. Line 2: wrap it in the CEILING(value, multiple of) function to bump up to next highest integer. Lines 3 and 4: same, but we use –NORM.S.INV(0.05) instead of the table lookup.

46. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8.8: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Mean (cont.) So the size of the sample that we need to construct a 90% confidence interval for the population mean with the desired margin of error is at least 48. What’s good about this? We have some assurance about the results we get, knowing how many we need in our sample. We know not to over-sample Saves time Saves effort Saves money! Drawback? Need to know σ. Or rely on some claim about σ.