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Section 8.1

Section 8.1. Estimating Population Means (When  is Known ). And some added content by D.R.S., University of Cordele. Estimating Population Means. A point estimate is a single-number estimate of a population parameter.

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Section 8.1

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  1. Section 8.1 Estimating Population Means (When  is Known) And some added contentby D.R.S., University of Cordele

  2. Estimating Population Means A point estimate is a single-number estimate of a population parameter. An unbiased estimatoris a point estimate that does not consistently underestimate or overestimate the population parameter.

  3. is the best point estimate of Usually, we can’t find the exact population mean by surveying the entire population (time, expense, or sheer impossibility). We can take a sample and found the . That’s as good as we can do. But we want to express some uncertainty. Being uncertain in a formal mathematical way.

  4. HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means Confidence Interval for 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. 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. 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. 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.”

  8. 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. 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. Example 8.2: Constructing a Confidence Interval with a Given Margin of Error (cont.) Thus, the lower endpoint is calculated as follows. Lower Endpoint of the Interval: = ______ - _______ = _______ hours per week Upper Endpoint of the Interval: = ______ - _______ = _______ hours per week Write it as an inequality: _______ < μ < ________ Write it as an interval: ( ______ , ______ )

  11. 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.”

  12. 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

  13. E = margin of errorThe parts of the formula c = the level of confidence, such as c = .90 for 90% alpha, α = 1 – c, such as 0.10 = the critical value, the z value such that the area to the right of is the area to the left of is the area in the middle is c σ = the population standard deviation n = the sample size

  14. A convenient table for common critical values You already know how to get these values. For example: For an 80% level of confidence, α = 1 – 80% = 1 – 0.8000 = 0.2000 α / 2 = 0.1000 What z (and –z) value causes area 0.8000 in the middle and 0.1000 in each tail?

  15. 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. Make notes as you read: 100 = _______ 15.50 = ______ 0.99 = _______ so _____ = ______

  16. Example 8.3: Finding the Margin of Error of a Confidence Interval for a Population Mean ( Known) Calculating by hand: So the confidence interval is ( ______, ______ ). Conclusion: “We are _____% confident that the mean ________________ is between $________ and $________.” σ = ______ c = ______, α = _____, α/2 = _____, z = _________ Sample: _______, n = ________. Plug in and compute to get E = _______. This problem only gave enough information for finding the Margin Of Error. It’s not a complete “Find the confidence interval” question. Those kinds are next. . .

  17. 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. Make notes as you read: 35 = _______ 42.0 = ______ 0.95 = _______ so _____ = ______ 6.5 = _______

  18. Example 8.4 – Suicide hotline planning Calculating by hand: So the confidence interval is ( ______, ______ ). Conclusion: “We are _____% confident that the mean _______________________ is between ________and ________.” σ = ______ c = ______, α = _____, α/2 = _____, z = _________ Sample: _______, n = ________. Plug in and compute to get E = _______.

  19. Example 8.4 – Suicide hotline planning Calculating with TI-84 ZInterval: (STAT, TESTS, 7) So the confidence interval is ( ______, ______ ). Conclusion: “We are _____% confident that the mean ________________ is between ________ and ________.”

  20. 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 collect data from the parents of 184 randomly selected children. The sample mean is found to be 4.7 toys per child. Construct a 99% 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. Make notes as you read: 184 = ______, 4.7 = _____, 99% = ______, 1.9 = ______

  21. Example 8.5 – Marketing research Calculating by hand: So the confidence interval is ( ______, ______ ). Conclusion: “We are _____% confident that the mean _______________________ is between ________and ________.” σ = ______ c = ______, α = _____, α/2 = _____, z = _________ Sample: _______, n = ________. Plug in and compute to get E = _______.

  22. Example 8.5 – Marketing research Calculating with TI-84 ZInterval: (STAT, TESTS, 7) So the confidence interval is ( ______, ______ ). Conclusion: “We are _____% confident that the mean ________________ is between ________ and ________.”

  23. Example 8.6: 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. Make notes as you read: 431 = ______, 8.72 = _____, 95% = ______, 0.36 = ______

  24. Example 8.6 – Marketing research - feet Calculating by hand: So the confidence interval is ( ______, ______ ). Conclusion: “We are _____% confident that the mean _______________________ is between ________and ________.” σ = ______ c = ______, α = _____, α/2 = _____, z = _________ Sample: _______, n = ________. Plug in and compute to get E = _______.

  25. Example 8.6 – Marketing research - feet Calculating with TI-84 ZInterval: (STAT, TESTS, 7) So the confidence interval is ( ______, ______ ). Conclusion: “We are _____% confident that the mean ________________ is between ________ and ________.”

  26. 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

  27. 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

  28. Minimum Sample Size for Estimating a Population Mean 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

  29. What is the Minimum Sample Size I need? The Situation: I’m going to calculatea 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.

  30. 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 zα/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.)

  31. 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.) ________, σ = ________, E = ________

  33. Example 8.8 with TI-84 – there is no special function.So you have to use the formula. 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 X2 key) Or even more slick: : (-invNorm(0.05)*8.4/2)2

  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. 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.

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