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Chapter 6 Confidence Intervals § 6.1 Confidence Intervals for the Mean (Large Samples) Point Estimate for Population μ A point estimate is a single value estimate for a population parameter. The most unbiased point estimate of the population mean, , is the sample mean, . Example: A random sample of 32 textbook prices (rounded to the nearest dollar) is taken from a local college bookstore. Find a point estimate for the population mean, . 34 56 79 94 34 65 86 95 38 65 87 96 45 66 87 98 45 67 87 98 45 67 88 101 45 68 90 110 54 74 90 121 74.22 The point estimate for the population mean of textbooks in the bookstore is $74.22. Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Interval Estimate An interval estimate is an interval, or range of values, used to estimate a population parameter. Point estimate for textbooks • 74.22 interval estimate How confident do we want to be that the interval estimate contains the population mean, μ? Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 Level of Confidence The level of confidence c is the probability that the interval estimate contains the population parameter. c (1 – c) c is the area beneath the normal curve between the critical values. (1 – c) zc z=0 z zc Critical values Use the Standard Normal Table to find the corresponding z-scores. The remaining area in the tails is 1 – c . Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Common Levels of Confidence If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population mean, μ. 0.90 0.05 0.05 zc = z1.645 z = 0 zc = z1.645 c c z The corresponding z-scores are ± 1.645. Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Common Levels of Confidence If the level of confidence is 95%, this means that we are 95% confident that the interval contains the population mean, μ. 0.95 0.025 0.025 zc = z1.96 c z=0 zc =z1.96 c z The corresponding z-scores are ± 1.96. Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 Common Levels of Confidence If the level of confidence is 99%, this means that we are 99% confident that the interval contains the population mean, μ. 0.99 0.005 0.005 zc = z2.575 z = 0 zc = z2.575 c c z The corresponding z-scores are ± 2.575. Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Margin of Error The difference between the point estimate and the actual population parameter value is called the sampling error. When μ is estimated, the sampling error is the difference μ – . Since μ is usually unknown, the maximum value for the error can be calculated using the level of confidence. Given a level of confidence, the margin of error (sometimes called the maximum error of estimate or error tolerance) E is the greatest possible distance between the point estimate and the value of the parameter it is estimating. E z cσ x z c σ n When n 30, the sample standard deviation, s, can be used for . Larson & Farber, Elementary Statistics: Picturing the World, 3e 9 Margin of Error Example: A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample is = 74.22, and the sample standard deviation is s = 23.44. Use a 95% confidence level and find the margin of error for the mean price of all textbooks in the bookstore. E z c σ 1.96 23.44 n 32 Since n 30, s can be substituted for σ. 8.12 We are 95% confident that the margin of error for the population mean (all the textbooks in the bookstore) is about $8.12. Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 Confidence Intervals for μ A c-confidence interval for the population mean μ is E<μ< + E. The probability that the confidence interval contains μ is c. Example: A random sample of 32 textbook prices is taken from a local college bookstore. The mean of the sample is = 74.22, the sample standard deviation is s = 23.44, and the margin of error is E = 8.12. Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 11 Confidence Intervals for μ Example continued: Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. = 74.22 s = 23.44 E = 8.12 Left endpoint = ? • E = 74.22 – 8.12 = 66.1 Right endpoint = ? =•74.22 • + E = 74.22 + 8.12 = 82.34 With 95% confidence we can say that the cost for all textbooks in the bookstore is between $66.10 and $82.34. Larson & Farber, Elementary Statistics: Picturing the World, 3e 12 Finding Confidence Intervals for μ Finding a Confidence Interval for a Population Mean 30 or σ known with a normally distributed population) In Words 1. Find the sample statistics n and (n In Symbols . x x n 2. Specify , if known. Otherwise, if n 30, find the sample standard deviation s and use it as an estimate for . ( x x )2 s n 1 3. Find the critical value zc that corresponds to the given level of confidence. Use the Standard Normal Table. E zc σ 4. Find the margin of error E. n Left endpoint: E 5. Find the left and right endpoints and Right endpoint: +E form the confidence interval. Interval: E < μ < +E Larson & Farber, Elementary Statistics: Picturing the World, 3e 13 Confidence Intervals for μ ( Known) Example: A random sample of 25 students had a grade point average with a mean of 2.86. Past studies have shown that the standard deviation is 0.15 and the population is normally distributed. Construct a 90% confidence interval for the population mean grade point average. = 2.86 = 0.15 n = 25 σ 1.645 0.15 0.05 zc = 1.645 E z c n 25 2.81 < μ < 2.91 ± E = 2.86 ± 0.05 With 90% confidence we can say that the mean grade point average for all students in the population is between 2.81 and 2.91. Larson & Farber, Elementary Statistics: Picturing the World, 3e 14 Sample Size Given a c-confidence level and a maximum error of estimate, E, the minimum sample size n, needed to estimate , the population mean, is zc n . E 2 If is unknown, you can estimate it using s provided you have a preliminary sample with at least 30 members. Example: You want to estimate the mean price of all the textbooks in the college bookstore. How many books must be included in your sample if you want to be 99% confident that the sample mean is within $5 of the population mean? Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 15 Sample Size Example continued: You want to estimate the mean price of all the textbooks in the college bookstore. How many books must be included in your sample if you want to be 99% confident that the sample mean is within $5 of the population mean? s = 23.44 = 74.22 zc = 2.575 zc 2.575 23.44 n E 5 145.7 (Always round up.) 2 2 You should include at least 146 books in your sample. Larson & Farber, Elementary Statistics: Picturing the World, 3e 16