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Chapter 6 Confidence Intervals Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 1 Chapter Outline • • • • . 6.1 Confidence Intervals for the Mean ( Known) 6.2 Confidence Intervals for the Mean ( Unknown) 6.3 Confidence Intervals for Population Proportions 6.4 Confidence Intervals for Variance and Standard Deviation Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 2 Section 6.2 Confidence Intervals for the Mean ( Unknown) . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 3 Section 6.2 Objectives • How to interpret the t-distribution and use a tdistribution table • How to construct and interpret confidence intervals for a population mean when is not known . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 4 The t-Distribution • When the population standard deviation is unknown, the sample size is less than 30, and the random variable x is approximately normally distributed, it follows a t-distribution. x - t s n • Critical values of t are denoted by tc. . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 5 Properties of the t-Distribution 1. The mean, median, and mode of the t-distribution are equal to zero. 2. The t-distribution is bell shaped and symmetric about the mean. 3. The total area under a t-curve is 1 or 100%. 4. The tails in the t-distribution are “thicker” than those in the standard normal distribution. 5. The standard deviation of the t-distribution varies with the sample size, but it is greater than 1. . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 6 Properties of the t-Distribution 6. 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 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 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 7 Properties of the t-Distribution 7. As the degrees of freedom increase, the tdistribution approaches the normal distribution. After 30 d.f., the t-distribution is very close to the standard normal z-distribution. d.f. = 2 d.f. = 5 Standard normal curve . t 0 Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 8 Example: Finding Critical Values of t Find the critical value tc for a 95% confidence when the sample size is 15. Solution: d.f. = n – 1 = 15 – 1 = 14 Table 5: t-Distribution tc = 2.145 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 9 Solution: Critical Values of t 95% of the area under the t-distribution curve with 14 degrees of freedom lies between t = +2.145. c = 0.95 t -tc = -2.145 . tc = 2.145 Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 10 Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ s • x E x E where E tc n • The probability that the confidence interval contains μ is c. . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 11 Confidence Intervals and t-Distributions In Words In Symbols 1. Verify that is not known, the sample is random, and the population is normally distributed or n 30. 2. Identify the sample statistics n, x , and s. . x x n Copyright © 2015, 2012, and 2009 Pearson Education, Inc. (x x )2 s n 1 12 Confidence Intervals and t-Distributions In Words In Symbols 3. Identify the degrees of freedom, the level of confidence c, and the critical value tc. d.f. = n – 1; Use Table 5. 4. Find the margin of error E. E tc 5. Find the left and right endpoints and form the confidence interval. . s n Left endpoint: x E Right endpoint: x E Interval: x E x E Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 13 Example: Constructing a Confidence Interval You randomly select 16 coffee shops and measure the temperature of the coffee sold at each. The sample mean temperature is 162.0ºF with a sample standard deviation of 10.0ºF. Find the 95% confidence interval for the mean temperature. Assume the temperatures are approximately normally distributed. Solution: Use the t-distribution (n < 30, σ is unknown, temperatures are approximately distributed.) . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 14 Solution: Constructing a Confidence Interval • n =16, x = 162.0 s = 10.0 c = 0.95 • df = n – 1 = 16 – 1 = 15 • Critical Value Table 5: t-Distribution tc = 2.131 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 15 Solution: Constructing a Confidence Interval • Margin of error: s 10 E tc 2.131 5.3 n 16 • Confidence interval: Left Endpoint: x E 162 5.3 156.7 Right Endpoint: xE 162 5.3 167.3 156.7 < μ < 167.3 . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 16 Solution: Constructing a Confidence Interval • 156.7 < μ < 167.3 Point estimate 156.7 ( x E 162.0 •x 167.3 ) xE With 95% confidence, you can say that the mean temperature of coffee sold is between 156.7ºF and 167.3ºF. . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 17 Normal or t-Distribution? . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 18 Example: Normal or t-Distribution? You randomly select 25 newly constructed houses. The sample mean construction cost is $181,000 and the population standard deviation is $28,000. Assuming construction costs are normally distributed, should you use the normal distribution, the t-distribution, or neither to construct a 95% confidence interval for the population mean construction cost? Solution: Use the normal distribution (the population is normally distributed and the population standard deviation is known) . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 19 Section 6.2 Summary • Interpreted the t-distribution and used a t-distribution table • Constructed and interpreted confidence intervals for a population mean when is not known . Copyright © 2015, 2012, and 2009 Pearson Education, Inc. 20