1. 1 BA 275 Quantitative Business Methods Statistical Inference: Confidence Interval Estimation Introduction Estimating the population mean  Examples Office Hour tomorrow (9:00 – 10:00, 2/7/06) has been re-scheduled to 1:00 – 3:00, 2/7/06. Agenda

2. 2 Midterm Examination #1

3. 3 Midterm #1 – Part 1 A survey is done to estimate how much time American college students spend using the Internet. Using random selection methods, 400 students are surveyed. Based on the results of the survey, it is estimated that college students use the Internet an average of 17.3 hours per week. Population of interest? Sample statistic? Value? Parameter of interest? Value?

4. 4 Midterm #1 – Part II Normal Distribution with two parameters  and  2.parameters  and  2 Given X ~ N(0, 4) and P(-a < X < a) = 0.874, find the value of a. Answer: 3.06 Let X ~ N(2, 1). Given that P(X b) = 0.05, what are the values of a and b? Answer: a = 0.718 and b = 3.645

5. 5 Midterm #1 – Part III New median = 346.371

6. 6 Midterm #1 – Part 4 -10 0 10 20 30 40 50

7. 7 Midterm #1 – Part 5

8. 8 Project 2 Project 1 Focus on sample and the use of Statgraphics. Project 2 Focus on extrapolating sample to the whole population. Need to provide in-depth analysis. Be thorough. What do I mean?

9. 9 Are birth rates related to death rates in the 91 countries? Not good enough for the 2 nd project.

10. 10 Are birth rates related to death rates in the 91 countries? Better but need more.

11. 11 Are birth rates related to death rates in the 91 countries? Even Better but can we do more? Mexico Avg 11931 21672 318706 47392 52332 6852

12. 12 Sampling Distribution (CyberStats) A sampling distribution describes the distribution of all possible values of a statistic over all possible random samples of a specific size that can be taken from a population.

13. 13 Central Limit Theorem (CLT) The CLT applied to Means

14. 14 Review Example The scores of students on the ACT college entrance examination in a recent year had a mean 18.6 and a standard deviation 5.9. What is the probability that a single student randomly chosen from all those taking the test scores 21 or higher? What is the probability that the mean score of 50 randomly selected students is 21 or higher?

15. 15 Statistical Inference: Estimation Research Question: What is the parameter value? Sample of size n Population Tools (i.e., formulas): Point Estimator Interval Estimator Example:  = 15 n = 400 What is the value of  ? Example:  ?

16. 16 Example 1 (p.14) The number of cars sold annually by used car salespeople is normally distributed with a standard deviation of 15. A random sample of 400 salespeople was selected. Find the 95% confidence interval estimate of the population mean. Interpret the interval estimate. The mean number of cars sold annually was found to be 75. How do you use the information provided to make an educated guess about the population mean? Is this condition necessary?

17. 17 100(1-  )% Confidence Interval for the Mean (p.13, formulas 1 and 2) If n is large If n is small

18. 18 Practice Problems (Z  /2 ) What are the values of z  /2 for 86%, 92%, and 97% confidence intervals? Which of the three intervals is wider?

19. 19 Example 1 (p.14) The number of cars sold annually by used car salespeople is normally distributed with a standard deviation of 15. A random sample of 400 salespeople was taken and the mean number of cars sold annually was found to be 75. Find the 90% confidence interval estimate of the population mean. Interpret the interval estimate. Z  /2=? (Formula (1) on page 13)

20. 20 Example 2 (p.14) Suppose that the amount of time teenagers spend weekly at part-time jobs is normally distributed with a standard deviation of 20 minutes. A random sample of 100 observations is drawn and the sample mean is computed as 125 minutes. Determine the 92% confidence interval estimate of the population mean.