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 (When  is Known) Some publisher content + vast improvements added by D.R.S., University of Cordele A point estimate is a single-number estimate of a population parameter (such as the population mean, the population standard deviation, the population variance). In 8.1 we estimate population means. An unbiased estimator is a point estimate that does not consistently underestimate or overestimate the population parameter.

3. 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

4. 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 SOLUTION:

5. 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. A point estimate is a single-number estimate of a population parameter. “We estimate that the population mean is 81.6.” “We are 90% confident that the population mean is between 77.5 and 85.8.” taking the estimation idea one step further

6. 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. (The sample mean plus or minus some “margin of error”.)

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.)

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.) The interpretation of our confidence interval is that we are ____% confident that the true population mean for the number of hours per week that students on this campus spend studying is between _____ and _____ 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.”

13. 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

14. E = margin of error The parts of the formula

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice/Review – Find the Area

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 So you can use this table with the special values. Or you can reason it out like you did on the previous slide where we calculated the critical value for a 90% confidence interval.

17. EXAMPLE: A sample of 32 airport shuttle vans’ records shows that the mean annual mileage was 75,118. Assume that the standard deviation of the population of all such vans is 10,000 miles. Construct the 90% confidence interval of the annual mileage. Construct the 95% confidence interval, too.

18. The next slide shows the final result of the by-hand solution. The next slide after that shows the final result of the TI-84 ZInterval solution. To view the entire step-by-step solution, go to http://2205.drscompany.com, click on Examples, click on Chapter 8, click on Section 8.1, then click “Confidence interval with z”.

19. Critical Value: z α/2 : z α/2 =1.645 for 90% C.I. z α/2 = 1.96 for 95% C.I. Margin of Error, E: For 90% C.I.: For 95% C.I.: A sample of 32 …mean annual mileage was 75,118…. standard deviation of the population of all such vans is 10000 miles. Construct the 90% and the 95% confidence intervals… stats/confint/z/01-basic 19 (1.645)(10000) √ 32 (1.96)(10000) √ 32 E = E = 2908.0 miles E = 3464.8 miles x – E = 75118 – 2908.0 = 72210.0 x + E = 75118 + 2908.0 = 78026.0 90% C.I. is (72210.0, 78026.0) miles or 72210.0 < μ < 78026.0 x – E = 75118 – 3464.8 = 71653.2 x + E = 75118 + 3464.8 = 78582.8 95% C.I. is (71653.2, 78582.8) miles or 71653.2 < μ < 78582.8 x = 75118 σ = 10000 n = 32

20. A sample of 32 …mean annual mileage was 75,118…. standard deviation of the population of all such vans is 10000 miles. Construct the 90% and the 95% confidence intervals… stats/confint/z/01-basic 20 90% C.I. is (72210.0, 78026.0) miles or 72210 < μ < 78026 95% C.I. is (71653, 78583) miles or 71653 < μ < 78583 x = 75118 σ = 10000 n = 32

21. TRY THIS: Suppose that on Thursday afternoon we had 40 applicants to the college take a placement exam. The mean score was a 38. The standard deviation of that exam score is 5.5 according to a recent study. Construct the 90% confidence interval for the population mean score of all applicants taking that exam. Make notes of values and variables: 40 = _______ 38 = _______ 5.5 = ______ 0.90 = ______

22. Compute the margin of error using the primitive formula: Then assemble the Confidence Interval:

23. Do it again using TI-84 ZInterval Inpt: Stats σ: ______________ x: ______________ n: ______________ C-Level: _______ Calculate What does the results screen tell you?

24. Further words about ZInterval If you’re asked for a confidence interval, Use ZInterval for a normal distribution 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

25. Excel Function =CONFIDENCE.NORM( α, σ, n ) α = level of significance = 1 – level of confidence σ = population standard deviation n = sample size (These values came from some other example)

26. 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

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. What is the Minimum Sample Size I need? The Situation: I’m going to calculate a confidence interval. I want to be within E =.... of the true mean. I want c =.... % confidence level. My Big Question: “What’s the minimum sample size I need to achieve this level of accuracy?” Why do I care? It costs time and money to collect the sample data.

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. What is the Minimum Sample Size I need? The Formula: I want to be within E =.... of the true mean. It goes right in. I want c =....% confidence level. Just like before, c leads to α leads to α/2 leads to z α/2. I have σ available from somewhere, probably other people’s previous studies. My Big Question: “What’s the minimum sample size I need to achieve this level of accuracy?” The formula tells me n. Always bump up (don’t round, bump up.)

29. EXAMPLE: The number of samples needed to estimate the price of something within $200. Go to http://2205.drscompany.com Click on Chapter 8 Click on Section 8.1 Click on What sample size is needed? The final results screen is shown in the next slide.

30. .  Recognize the problem type.  Write the formula.  Plug into the formula.  Calculator.  Always bump up to the next higher whole number. Do not round off in this calculation; always bump up. 95% confidence interval …The standard deviation … assumed to be $1,000. If the maximum allowable margin of error is $200, how many prices need[ed] … for the sample? stats/confint/z/05-sample-size-needed 30 z a/2 σ E n = ( ) 2 (1.96)(1000) 200 n = ( ) 2 n = 97 is the minimum sample size needed to obtain a 95% confidence interval with a margin of error of no more than $200.

31. 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. Make notes as you read: 90% = ______ 2 = _____, 8.4 = ______.

32. Example 8.8: Finding the Minimum Sample Size Needed for a Confidence Interval for a Population Mean (cont.)

33. 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 _______. 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.

34. 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.

35. One more kind of problem you’ll see Example: A survey showed that the 95% confidence interval for the price of homes in Pine City is ($101000, $145000). (a)What was the sample mean? (b)What is the point estimate for the population mean? (c)What is the margin of error? (d)What is the standard deviation? http://2205.drscompany.com, Chapter 8, Section 8.1, “Working backward from the interval”