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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 x. 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 x 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 (1 – c) 2 c is the area beneath the normal curve between the critical values. 1 (1 – c) 2 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 – x . 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 x = 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 xE xE 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 x = 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. x = 74.22 s = 23.44 E = 8.12 Left endpoint = ? • x E 74.22 8.12 = 66.1 Right endpoint = ? x =• 74.22 • x 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 (n In Symbols 1. Find the sample statistics n and x . 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. 4. Find the margin of error E. 5. Find the left and right endpoints and form the confidence interval. E zc σ n Left endpoint: x E Right endpoint: x E Interval: x E x 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. x = 2.86 n = 25 = 0.15 σ 1.645 0.15 0.05 zc = 1.645 E z c n 25 x E 2.86 ± 0.05 2.81 < μ < 2.91 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? x = 74.22 s = 23.44 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 § 6.2 Confidence Intervals for the Mean (Small Samples) The t-Distribution When a sample size is less than 30, and the random variable x is approximately normally distributed, it follow a t-distribution. t x μ s n Properties of the t-distribution 1. The t-distribution is bell shaped and symmetric about the mean. 2. The t-distribution is a family of curves, each determined by a parameter called the degrees of freedom. The degrees of freedom are the number of free choices left after a sample statistic such as x is calculated. When you use a t-distribution to estimate a population mean, the degrees of freedom are equal to one less than the sample size. d.f. = n – 1 Degrees of freedom Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 18 The t-Distribution 3. The total area under a t-curve is 1 or 100%. 4. The mean, median, and mode of the t-distribution are equal to zero. 5. As the degrees of freedom increase, the t-distribution approaches the normal distribution. After 30 d.f., the t-distribution is very close to the standard normal z-distribution. The tails in the t-distribution are “thicker” than those in the standard normal distribution. d.f. = 2 d.f. = 5 0 t Standard normal curve Larson & Farber, Elementary Statistics: Picturing the World, 3e 19 Critical Values of t Example: Find the critical value tc for a 95% confidence when the sample size is 5. Appendix B: Table 5: t-Distribution Level of confidence, c One tail, d.f. Two tails, 1 2 3 4 5 0.50 0.25 0.50 1.000 .816 .765 .741 .727 0.80 0.10 0.20 3.078 1.886 1.638 1.533 1.476 0.90 0.95 0.98 0.05 0.025 0.01 0.10 0.05 0.02 6.314 12.706 31.821 2.920 4.303 6.965 2.353 3.182 4.541 2.132 2.776 3.747 2.015 2.571 3.365 d.f. = n – 1 = 5 – 1 = 4 tc = 2.776 c = 0.95 Larson & Farber, Elementary Statistics: Picturing the World, 3e Continued. 20 Critical Values of t Example continued: Find the critical value tc for a 95% confidence when the sample size is 5. 95% of the area under the t-distribution curve with 4 degrees of freedom lies between t = ±2.776. c = 0.95 tc = 2.776 tc = 2.776 Larson & Farber, Elementary Statistics: Picturing the World, 3e t 21 Confidence Intervals and t-Distributions Constructing a Confidence Interval for the Mean: Distribution In Words t- In Symbols 1. Identify the sample statistics n, x, and s. 2. Identify the degrees of freedom, the level of confidence c, and the critical value tc. 3. Find the margin of error E. 4. Find the left and right endpoints and form the confidence interval. x x n ( x x )2 s n 1 d.f. = n – 1 E tc s n Left endpoint: x E Right endpoint: x E Interval: x E x E Larson & Farber, Elementary Statistics: Picturing the World, 3e 22 Constructing a Confidence Interval Example: In a random sample of 20 customers at a local fast food restaurant, the mean waiting time to order is 95 seconds, and the standard deviation is 21 seconds. Assume the wait times are normally distributed and construct a 90% confidence interval for the mean wait time of all customers. n = 20 x 95 d.f. = 19 tc = 1.729 x E = 95 ± 8.1 s = 21 21 8.1 E tc s 1.729 n 20 86.9 < μ < 103.1 We are 90% confident that the mean wait time for all customers is between 86.9 and 103.1 seconds. Larson & Farber, Elementary Statistics: Picturing the World, 3e 23 Normal or t-Distribution? Use the normal distribution with Is n 30? Yes E zc σ . n If is unknown, use s instead. No Is the population normally, or approximately normally, distributed? No Yes You cannot use the normal distribution or the t-distribution. Use the normal distribution with Is known? No Yes E zc σ . n Use the t-distribution with E tc s n and n – 1 degrees of freedom. Larson & Farber, Elementary Statistics: Picturing the World, 3e 24 Normal or t-Distribution? Example: Determine whether to use the normal distribution, the t-distribution, or neither. a.) n = 50, the distribution is skewed, s = 2.5 The normal distribution would be used because the sample size is 50. b.) n = 25, the distribution is skewed, s = 52.9 Neither distribution would be used because n < 30 and the distribution is skewed. c.) n = 25, the distribution is normal, = 4.12 The normal distribution would be used because although n < 30, the population standard deviation is known. Larson & Farber, Elementary Statistics: Picturing the World, 3e 25 § 6.3 Confidence Intervals for Population Proportions Point Estimate for Population p The probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by pˆ x n where x is the number of successes in the sample and n is the number in the sample. The point estimate for the proportion of failures is qˆ = 1 – p̂. The symbols p̂ and qˆ are read as “p hat” and “q hat.” Larson & Farber, Elementary Statistics: Picturing the World, 3e 27 Point Estimate for Population p Example: In a survey of 1250 US adults, 450 of them said that their favorite sport to watch is baseball. Find a point estimate for the population proportion of US adults who say their favorite sport to watch is baseball. n = 1250 x = 450 pˆ x 450 0.36 n 1250 The point estimate for the proportion of US adults who say baseball is their favorite sport to watch is 0.36, or 36%. Larson & Farber, Elementary Statistics: Picturing the World, 3e 28 Confidence Intervals for p A c-confidence interval for the population proportion p is pˆ E p pˆ E where E zc pq ˆ ˆ. n The probability that the confidence interval contains p is c. Example: Construct a 90% confidence interval for the proportion of US adults who say baseball is their favorite sport to watch. n = 1250 x = 450 p̂ 0.36 Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 29 Confidence Intervals for p Example continued: n = 1250 p̂ 0.36 x = 450 ˆˆ E z c pq n qˆ 0.64 Left endpoint = ? • p̂ E 0.36 0.022 0.338 1.645 (0.36)(0.64) 0.022 1250 Right endpoint = ? p̂ • 0.36 • p̂ E 0.36 0.022 0.382 With 90% confidence we can say that the proportion of all US adults who say baseball is their favorite sport to watch is between 33.8% and 38.2%. Larson & Farber, Elementary Statistics: Picturing the World, 3e 30 Finding Confidence Intervals for p Constructing a Confidence Interval for a Population Proportion In Words In Symbols 1. Identify the sample statistics n and x. 2. Find the point estimate p̂. 3. Verify that the sampling distribution can be approximated by the normal distribution. 4. Find the critical value zc that corresponds to the given level of confidence. 5. Find the margin of error E. 6. Find the left and right endpoints and form the confidence interval. pˆ x n npˆ 5, nqˆ 5 Use the Standard Normal Table. ˆˆ E z c pq n Left endpoint: p̂ E Right endpoint: p̂ E Interval: pˆ E p pˆ E Larson & Farber, Elementary Statistics: Picturing the World, 3e 31 Sample Size Given a c-confidence level and a margin of error, E, the minimum sample size n, needed to estimate p is 2 zc ˆˆ . n pq E This formula assumes you have an estimate for p̂ and qˆ. If not, use pˆ 0.5 and qˆ 0.5. Example: You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 32 Sample Size Example continued: You wish to find out, with 95% confidence and within 2% of the true population, the proportion of US adults who say that baseball is their favorite sport to watch. n = 1250 x = 450 2 p̂ 0.36 1.96 z ˆ ˆ c (0.36)(0.64) n pq 0.02 E 2 2212.8 (Always round up.) You should sample at least 2213 adults to be 95% confident. Larson & Farber, Elementary Statistics: Picturing the World, 3e 33 § 6.4 Confidence Intervals for Variance and Standard Deviation The Chi-Square Distribution The point estimate for 2 is s2, and the point estimate for is s. s2 is the most unbiased estimate for 2. You can use the chi-square distribution to construct a confidence interval for the variance and standard deviation. If the random variable x has a normal distribution, then the distribution of 2 (n 1)s 2 σ2 forms a chi-square distribution for samples of any size n > 1. Larson & Farber, Elementary Statistics: Picturing the World, 3e 35 The Chi-Square Distribution Four properties of the chi-square distribution are as follows. 1. All chi-square values χ2 are greater than or equal to zero. 2. The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for 2, use the χ2-distribution with degrees of freedom. To form a confidence interval for 2, use the χ2distribution with degrees of freedom equal to one less than the sample size. 3. The area under each curve of the chi-square distribution equals one. 4. Find the critical value zc that corresponds to the given level of confidence. 5. Chi-square distributions are positively skewed. Larson & Farber, Elementary Statistics: Picturing the World, 3e 36 Critical Values for X2 There are two critical values for each level of confidence. The value χ2R represents the right-tail critical value and χ2L represents the left-tail critical value. 1 1 c 2 X2R X2 X2 X2L Area to the right of X2R 1 2 c 1 2 c Area to the right of X2L 1 c 2 X2L c 1 c 2 X2R The area between the left and right critical values is c. X2 Larson & Farber, Elementary Statistics: Picturing the World, 3e 37 Critical Values for X2 Example: Find the critical values χ2R and χ2L for an 80% confidence when the sample size is 18. Because the sample size is 18, there are d.f. = n – 1 = 18 – 1 = 17 degrees of freedom, Area to the right of χ2R = 1 c 1 0.8 0.1 2 2 Area to the right of χ2L = 1 c 1 0.8 0.9 2 2 Use the Chi-square distribution table to find the critical values. Larson & Farber, Elementary Statistics: Picturing the World, 3e Continued. 38 Critical Values for X2 Example continued: Appendix B: Table 6: χ2-Distribution Degrees of freedom 1 2 3 0.010 0.072 0.975 - 0.001 0.020 0.051 0.115 0.216 0.95 0.004 0.103 0.352 5.142 5.697 6.265 5.812 6.408 7.015 7.962 9.312 23.542 26.296 8.672 10.085 24.769 27.587 9.390 10.865 25.989 28.869 0.995 16 17 18 0.99 6.908 7.564 8.231 0.90 0.016 0.211 0.584 0.10 2.706 4.605 6.251 0.05 3.841 5.991 7.815 χ2R = 24.769 χ2L = 10.085 Larson & Farber, Elementary Statistics: Picturing the World, 3e 39 Confidence Intervals for 2 and A c-confidence interval for a population variance and standard deviation is as follows. Confidence Interval for 2: (n 1)s 2 2 XR 2 σ (n 1)s 2 X L2 Confidence Interval for : (n 1)s 2 2 XR σ (n 1)s 2 X L2 The probability that the confidence intervals contain 2 or is c. Larson & Farber, Elementary Statistics: Picturing the World, 3e 40 Confidence Intervals for 2 and Constructing a Confidence Interval for a Variance and a Standard Deviation In Words In Symbols 1. Verify that the population has a normal distribution. 2. Identify the sample statistic n and the degrees of freedom. d.f. = n 1 3. Find the point estimate s2. (x x )2 s n 1 4. Find the critical value χ2R and χ2L that correspond to the given level of confidence. 2 Use Table 6 in Appendix B. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 41 Confidence Intervals for 2 and Constructing a Confidence Interval for a Variance and a Standard Deviation In Words In Symbols 5. Find the left and right endpoints and form the confidence interval. 6. Find the confidence interval for the population standard deviation by taking the square root of each endpoint. (n 1)s 2 2 XR (n 1)s 2 2 XR Larson & Farber, Elementary Statistics: Picturing the World, 3e 2 σ σ (n 1)s 2 X L2 (n 1)s 2 X L2 42 Constructing a Confidence Interval Example: You randomly select and weigh 41 samples of 16-ounce bags of potato chips. The sample standard deviation is 0.05 ounce. Assuming the weights are normally distributed, construct a 90% confidence interval for the population standard deviation. d.f. = n – 1 = 41 – 1 = 40 degrees of freedom, Area to the right of χ2R = 1 c 1 0.9 0.05 2 2 Area to the right of χ2L = 1 c 1 0.9 0.95 2 2 The critical values are χ2R = 55.758 and χ2L = 26.509. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 43 Constructing a Confidence Interval Example continued: χ2L = 26.509 χ2R = 55.758 Left endpoint = ? (n 1)s 2 R2 Right endpoint = ? • (41 1)(0.05)2 55.758 0.04 (n 1)s 2 L2 • (41 1)(0.05)2 26.509 0.06 0.04 σ 0.06 With 90% confidence we can say that the population standard deviation is between 0.04 and 0.06 ounces. Larson & Farber, Elementary Statistics: Picturing the World, 3e 44