1. 1 Business 90: Business Statistics Professor David Mease Sec 03, T R 7:30-8:45AM BBC 204 Lecture 22 = More of Chapter “Confidence Interval Estimation” (CIE) Agenda: 1) Go over quiz over Homework 7 2) Reminder about Homework 8 (due Tuesday 5/4) 3) Lecture over more of Chapter CIE

2. 2 Homework 8 – Due Tuesday 5/4 1) Read chapter entitled “Confidence Interval Estimation” but only sections 1, 2 and 4 2) In that chapter do textbook problems 2, 12, 16, 32 and 42 3) A random sample of 8 SJSU students is taken. The ages of the students are 19,23,30,30,45,25,24,20 a. Using this data give a 90% confidence interval for the mean age of all SJSU students. Do NOT use Excel. b. If a sample of 8 students is taken every year, how often will the resulting 90% confidence intervals contain the true population mean? c. Are your answers to A and B correct even if the population of ages for the students is not normal? Why or why not? d. Use Excel to check your answer for Part A. 4) a. Everything else being equal, which will be narrower: a 90% confidence interval or a 95% confidence interval? b. Everything else being equal, which will be narrower: a confidence interval using a sample size of 1000 or a confidence interval using a sample size of 10,000? c. Everything else being equal, which will be narrower: a confidence interval where the standard deviation is 10 or a confidence interval where the standard deviation is 3?

3. 3 Confidence Interval Estimation Statistics for Managers Using Microsoft ® Excel 4 th Edition

4. 4 Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single population mean using both the normal and t distributions Determine the required sample size to estimate a mean or within a specified error

5. 5 Confidence Intervals Content of this chapter Confidence Intervals for the Population Mean, µ when Population Standard Deviation  is Known when Population Standard Deviation  is Unknown Determining the Required Sample Size

6. 6 Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

7. 7 Estimation Process (mean, µ, is unknown) Population Random Sample Mean X = 50 Sample I am 95% confident that µ is between 40 & 60.

8. 8 Confidence Interval Estimate An interval gives a range of values: Takes into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Gives information about closeness to unknown population parameters Stated in terms of level of confidence Can never be 100% confident

9. 9 In class exercise #88: Time spent using email per session is normally distributed with a mean of 8 minutes and a standard deviation of 2 minutes. If a random sample of 25 sessions is selected, answer the following. a) The sample mean will between what two values (symmetric about the mean) with 95% probability ? b) Complete the following sentence: “The sample mean will be within a distance of _______ minutes from the true mean 95% if the time.” c) Give a general formula for the answer in part b. d) Are your answers correct even if the population is not normal? Why or why not?

10. 10 Confidence Interval for μ (  Known) Assumptions Population standard deviation  is known Population is normally distributed If population is not normal, use central limit theorem if sample size is large (bigger than 30) Confidence interval estimate:

11. 11 In class exercise #89: (This is textbook problem 2 for your next homework) If the sample mean is 125, the standard deviation is 24 and the sample size is 36, construct a 99% confidence interval for the population mean.

12. 12 In class exercise #90: If the sample mean is 125, standard deviation is 24 and the sample size is 36, construct a 90% confidence interval for the population mean. How is this different from your answer for ICE #89. Why?

13. 13 In class exercise #91: If the sample mean is 125, standard deviation is 24 and the sample size is 200, construct a 90% confidence interval for the population mean. How is this different from your answer for ICE #90. Why?

14. 14 In class exercise #92: Remember the data for the 1500 houses in the file http://www.cob.sjsu.edu/mease_d/houses.xls from the 2nd homework. Considering these 1500 as the population, suppose we only had a sample consisting of the first five. a) Construct a 95% confidence interval for the population mean. (You can use Excel to get the standard deviation of all 1500 prices) b) Does your 95% confidence interval contain the true population mean? c) Will your 95% confidence interval always contain the true population mean? If not, how often?

15. 15 If the population standard deviation  is unknown, we can substitute the sample standard deviation, S This introduces extra uncertainty, since S is variable from sample to sample So we use the t distribution instead of the normal distribution (table 3) Confidence Interval for μ (σ Unknown)

16. 16 Assumptions Population standard deviation is unknown Population is normally distributed If population is not normal, use central limit theorem if sample size is large (bigger than 30) Use Student’s t Distribution Confidence Interval Estimate: Confidence Interval for μ (σ Unknown) (continued) t n-1 is found using table 3 with n-1 “Degrees of Freedom”

17. 17 Student’s t Distribution t 0 t (df = 5) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df =infinity ) Note: t Z as n increases

18. 18 In class exercise #93: If the sample mean is 125, the sample standard deviation is 24 and the sample size is 36, construct a 99% confidence interval for the population mean. Compare your answer here to your answer for ICE #89.