1. Confidence Intervals Chapter 6

2. § 6.1 Confidence Intervals for the Mean (Large Samples)

3. Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 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 3845 54 5665 6667 6874 798687 8890 94959698 101110121 The point estimate for the population mean of textbooks in the bookstore is $74.22.

4. Larson & Farber, Elementary Statistics: Picturing the World, 3e 4 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?

5. Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Level of Confidence The level of confidence c is the probability that the interval estimate contains the population parameter. z z = 0 zczc zczc Critical values c is the area beneath the normal curve between the critical values. The remaining area in the tails is 1 – c. c Use the Standard Normal Table to find the corresponding z - scores. (1 – c)

6. Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 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. z z = 0 zczc zczc The corresponding z - scores are ± 1.645. 0.90 0.05  z c =  1.645 z c = 1.645

7. Larson & Farber, Elementary Statistics: Picturing the World, 3e 7 z z = 0 zczc zczc 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. The corresponding z - scores are ± 1.96. 0.95 0.025  z c =  1.96 z c = 1.96

8. Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 z z = 0 zczc zczc 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. The corresponding z - scores are ± 2.575. 0.99 0.005  z c =  2.575 z c = 2.575

9. Larson & Farber, Elementary Statistics: Picturing the World, 3e 9 Margin of Error The difference between the point estimate and the actual population parameter value is called the sampling error. 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. When n  30, the sample standard deviation, s, can be used for . 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.

10. Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 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. We are 95% confident that the margin of error for the population mean (all the textbooks in the bookstore) is about $8.12. Since n  30, s can be substituted for σ.

11. Larson & Farber, Elementary Statistics: Picturing the World, 3e 11 Confidence Intervals for μ A c - confidence interval for the population mean μ is The probability that the confidence interval contains μ is c. Continued. 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.

12. Larson & Farber, Elementary Statistics: Picturing the World, 3e 12 Confidence Intervals for μ Example continued: Construct a 95% confidence interval for the mean price of all textbooks in the bookstore. s = 23.44E = 8.12 With 95% confidence we can say that the cost for all textbooks in the bookstore is between $66.10 and $82.34. Left endpoint = ?Right endpoint = ? = 66.1= 82.34

13. Larson & Farber, Elementary Statistics: Picturing the World, 3e 13 Finding Confidence Intervals for μ Finding a Confidence Interval for a Population Mean (n  30 or σ known with a normally distributed population) In Words In 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 . 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.

14. Larson & Farber, Elementary Statistics: Picturing the World, 3e 14 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. n = 25  = 0.15 z c = 1.645 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.

15. Larson & Farber, Elementary Statistics: Picturing the World, 3e 15 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 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.

16. Larson & Farber, Elementary Statistics: Picturing the World, 3e 16 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 z c = 2.575 You should include at least 146 books in your sample. (Always round up.)

17. § 6.2 Confidence Intervals for the Mean (Small Samples)

18. Larson & Farber, Elementary Statistics: Picturing the World, 3e 18 The t - Distribution 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 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. When a sample size is less than 30, and the random variable x is approximately normally distributed, it follow a t - distribution.

19. Larson & Farber, Elementary Statistics: Picturing the World, 3e 19 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. t 0 Standard normal curve The tails in the t - distribution are “thicker” than those in the standard normal distribution. d.f. = 5 d.f. = 2

20. Larson & Farber, Elementary Statistics: Picturing the World, 3e 20 Critical Values of t Example : Find the critical value t c for a 95% confidence when the sample size is 5. Appendix B: Table 5: t-Distribution Level of confidence, c 0.500.800.900.950.98 One tail,  0.250.100.050.0250.01 d.f. Two tails,  0.500.200.100.050.02 11.0003.0786.31412.70631.821 2.8161.8862.9204.3036.965 3.7651.6382.3533.1824.541 4.7411.5332.1322.7763.747 5.7271.4762.0152.5713.365 Continued. d.f. = n – 1 = 5 – 1 = 4 c = 0.95 t c = 2.776

21. Larson & Farber, Elementary Statistics: Picturing the World, 3e 21 Critical Values of t Example continued : Find the critical value t c 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. t  t c =  2.776 t c = 2.776 c = 0.95

22. Larson & Farber, Elementary Statistics: Picturing the World, 3e 22 Confidence Intervals and t - Distributions Constructing a Confidence Interval for the Mean: t- Distribution In Words In Symbols 1.Identify the sample statistics n, and s. 2.Identify the degrees of freedom, the level of confidence c, and the critical value t c. 3.Find the margin of error E. 4.Find the left and right endpoints and form the confidence interval. d.f. = n – 1

23. Larson & Farber, Elementary Statistics: Picturing the World, 3e 23 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. We are 90% confident that the mean wait time for all customers is between 86.9 and 103.1 seconds. s = 21 t c = 1.729 n = 20 d.f. = 19 86.9 < μ < 103.1

24. Larson & Farber, Elementary Statistics: Picturing the World, 3e 24 Normal or t-Distribution? Is n  30? Is the population normally, or approximately normally, distributed? You cannot use the normal distribution or the t-distribution. No Yes Is  known? No Use the normal distribution with If  is unknown, use s instead. YesNo Use the normal distribution with Yes Use the t - distribution with and n – 1 degrees of freedom.

25. Larson & Farber, Elementary Statistics: Picturing the World, 3e 25 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.

26. § 6.3 Confidence Intervals for Population Proportions

27. Larson & Farber, Elementary Statistics: Picturing the World, 3e 27 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 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 = 1 – The symbols and are read as “p hat” and “q hat.”

28. Larson & Farber, Elementary Statistics: Picturing the World, 3e 28 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. The point estimate for the proportion of US adults who say baseball is their favorite sport to watch is 0.36, or 36%. n = 1250x = 450

29. Larson & Farber, Elementary Statistics: Picturing the World, 3e 29 Confidence Intervals for p A c - confidence interval for the population proportion p is where 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. Continued. n = 1250x = 450

30. Larson & Farber, Elementary Statistics: Picturing the World, 3e 30 Confidence Intervals for p Example continued: 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%. Left endpoint = ?Right endpoint = ? n = 1250x = 450

31. Larson & Farber, Elementary Statistics: Picturing the World, 3e 31 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 3.Verify that the sampling distribution can be approximated by the normal distribution. 4.Find the critical value z c 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. Use the Standard Normal Table. Left endpoint : Right endpoint : Interval :