2. Chapter Outline 6.1 Confidence Intervals for the Mean (Large Samples) 6.2 Confidence Intervals for the Mean (Small Samples) 6.3 Confidence Intervals for Population Proportions Larson/Farber 4th ed2

3. Example: Point Estimate for Population μ Market researchers use the number of sentences per advertisement as a measure of readability for magazine advertisements. The following represents a random sample of the number of sentences found in 50 advertisements. Find a point estimate of the population mean, . (Source: Journal of Advertising Research) Larson/Farber 4th ed3 9 20 18 16 9 9 11 13 22 16 5 18 6 6 5 12 25 17 23 7 10 9 10 10 5 11 18 18 9 9 17 13 11 7 14 6 11 12 11 6 12 14 11 9 18 12 12 17 11 20

4. Solution: Point Estimate for Population μ Larson/Farber 4th ed4 The sample mean of the data is Your point estimate for the mean length of all magazine advertisements is 12.4 sentences.

5. Interval Estimate Interval estimate An interval, or range of values, used to estimate a population parameter. Larson/Farber 4th ed5 Point estimate 12.4 How confident do we want to be that the interval estimate contains the population mean μ? ( ) Interval estimate

6. Confidence Intervals for the Population Mean A c-confidence interval for the population mean μ The probability that the confidence interval contains μ is c. Larson/Farber 4th ed6

7. Solution: Finding the Margin of Error Larson/Farber 4th ed7 You don’t know σ, but since n ≥ 30, you can use s in place of σ. You are 95% confident that the margin of error for the population mean is about 1.4 sentences.

8. Example: Constructing a Confidence Interval Construct a 95% confidence interval for the mean number of sentences in all magazine advertisements. Larson/Farber 4th ed8 Solution: Recall and E = 1.4 11.0 < μ < 13.8 Left Endpoint:Right Endpoint:

9. Constructing Confidence Intervals for μ Larson/Farber 4th ed9 Finding a Confidence Interval for a Population Mean (n  30 or σ known with a normally distributed population) In WordsIn Symbols 1.Find the sample statistics n and. 2.Specify , if known. Otherwise, if n  30, find the sample standard deviation s and use it as an estimate for .

10. Constructing Confidence Intervals for μ Larson/Farber 4th ed10 3.Find the critical value z c that corresponds to the given level of confidence. 4.Find the margin of error E. 5.Find the left and right endpoints and form the confidence interval. Use the Standard Normal Table. Left endpoint: Right endpoint: Interval: In WordsIn Symbols

11. Interpreting the Results μ is a fixed number. It is either in the confidence interval or not. Incorrect: “There is a 90% probability that the actual mean is in the interval (22.3, 23.5).” Correct: “If a large number of samples is collected and a confidence interval is created for each sample, approximately 90% of these intervals will contain μ. Larson/Farber 4th ed11

12. Sample Size Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate the population mean  is If  is unknown, you can estimate it using s provided you have a preliminary sample with at least 30 members. Larson/Farber 4th ed12

13. Example: Sample Size You want to estimate the mean number of sentences in a magazine advertisement. How many magazine advertisements must be included in the sample if you want to be 95% confident that the sample mean is within one sentence of the population mean? Assume the sample standard deviation is about 5.0. Larson/Farber 4th ed13

14. Solution: Sample Size z c = 1.96   s = 5.0 E = 1 Larson/Farber 4th ed14 When necessary, round up to obtain a whole number. You should include at least 97 magazine advertisements in your sample.

15. Section 6.2 Confidence Intervals for the Mean (Small Samples) Larson/Farber 4th ed15

16. 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.. Larson/Farber 4th ed16

17. Confidence Intervals for the Population Mean σ unknown and n < 30 The probability that the confidence interval contains μ is c. Larson/Farber 4th ed17

18. Example: Critical Values of t Find the critical value t c for a 95% confidence when the sample size is 15. Larson/Farber 4th ed18 Table 5: t-Distribution t c = 2.145 Solution: d.f. = n – 1 = 15 – 1 = 14

19. 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. Larson/Farber 4th ed19 Solution: Use the t-distribution (n < 30, σ is unknown, temperatures are approximately distributed.)

20. Larson/Farber 4th ed20 n = 16 x bar = 162 s = 10 want a 95% confidence interval

21. Solution: Constructing a Confidence Interval n =16, x = 162.0 s = 10.0 c = 0.95 df = n – 1 = 16 – 1 = 15 Larson/Farber 4th ed21 Table 5: t-Distribution t c = 2.131

24. Section 6.3 Confidence Intervals for Population Proportions Larson/Farber 4th ed24

25. Section 6.3 Objectives Find a point estimate for the population proportion Construct a confidence interval for a population proportion Determine the minimum sample size required when estimating a population proportion Larson/Farber 4th ed25

26. Point Estimate for Population p Population Proportion The probability of success in a single trial of a binomial experiment. Denoted by p Point Estimate for p The proportion of successes in a sample. Denoted by – – read as “p hat” Larson/Farber 4th ed26

27. Example: Confidence Interval for p In a survey of 1219 U.S. adults, 354 said that their favorite sport to watch is football. Construct a 95% confidence interval for the proportion of adults in the United States who say that their favorite sport to watch is football. 354/1219 =.2904 Larson/Farber 4th ed27 So3543

28. Solution: Confidence Interval for p Larson/Farber 4th ed28 Margin of error:

29. Solution: Confidence Interval for p Confidence interval: Larson/Farber 4th ed29 Left Endpoint:Right Endpoint: 0.265 < p < 0.315

30. Solution: Confidence Interval for p 0.265 < p < 0.315 Larson/Farber 4th ed30 ( ) 0.29 0.2650.315 With 95% confidence, you can say that the proportion of adults who say football is their favorite sport is between 26.5% and 31.5%. Point estimate