1 Business 90: Business Statistics Professor David Mease Sec 03, T R 7:30-8:45AM BBC 204 Lecture 23 = Finish Chapter “Confidence Interval Estimation” (CIE) Agenda: 1) Reminder about Homework 8 (due Tuesday) 2) Announcement: Shortened office hours today 3) Lecture over rest of Chapter CIE

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 Announcement: Today office hours will end early at 12 noon instead of the usual time. If you have questions after 12 noon please or call ( ).

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

5 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

6 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

7 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

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

9 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

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 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)

12 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”

13 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

14 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.

15 In class exercise #94: (This is textbook problem 12 on your homework) If the sample mean is 50, the standard deviation is 15 and the sample size is 16, construct a 99% confidence interval for the population mean.

16 In class exercise #95: If the sample mean is 50, the standard deviation is 15 and the sample size is 16, construct a 95% confidence interval for the population mean. How and why is your answer different from your answer for ICE #94?

17 In class exercise #96: This exercise involves the data for the 1500 houses in the file from the 2nd homework. Considering these 1500 as the population, suppose we only had a sample consisting of the first five. Construct a 95% confidence interval for the population mean.

18 Excel will compute the “margin of error” for you using the raw data if you have the data analysis ToolPak installed Confidence Interval for μ (σ Unknown) using Excel

19 In class exercise #97: (This is like #3 on your homework) A random sample of 4 employees from a company is taken to determine the mean salary for the entire company. Their salaries (in thousands) are listed here : A) Give a point estimate for the mean salary in the company. B) Construct a 90% confidence interval for the mean salary in the company by hand. C) Is your answer in B correct if the population does not have a normal distribution? Why or why not? D) Check your answers in A and B using excel.

20 Determining Sample Size (Remember to always round up to the nearest whole number when using this formula.)

21 In class exercise #98: A consumer group wants to estimate the average amount of money spent per household for combined cable television and internet service each month. Based on prior data they believe the standard deviation is approximately $20. What is the sample size needed to get a margin of error of $2 with 99% confidence?

22 In class exercise #99: (Part A is textbook problem 32 on your homework) A) If you want to be 95% confident of estimating the population mean to within a sampling error of ±5 and the standard deviation is assumed to be equal to 15, what sample size is required? B) If a sample of this size is collected and the sample mean is 100, construct the 95% confidence interval for the population mean.