1. 1 Business 90: Business Statistics Professor David Mease Sec 03, T R 7:30-8:45AM BBC 204 Lecture 20 = Finish Chapter “The Normal Distribution and Other Continuous Distributions” (TNDAOCD) Agenda: 1) Reminder about Homework 7 (due Thursday) 2) Lecture over rest of Chapter TNDAOCD

2. 2 1) Read chapter entitled “The Normal Distribution and Other Continuous Distributions” but only sections 1, 5 and 6. 2) In that chapter do textbook problems 6, 8, 38 and 44. Homework 7 – Due Thursday 4/22

3. 3 The Normal Distribution and Other Continuous Distributions Statistics for Managers Using Microsoft ® Excel 4 th Edition

4. 4 Chapter Goals After completing this chapter, you should be able to: Describe the characteristics of the normal distribution Translate normal distribution problems into standardized normal distribution problems Find probabilities using a normal distribution table Define the concept of a sampling distribution Determine the mean and standard deviation for the sampling distribution of the sample mean Describe the Central Limit Theorem and its importance Apply the sampling distribution for the sample mean

5. 5 Probability Distributions Random Variable Represents a possible numerical value from an uncertain event Random Variables Discrete Random Variable Continuous Random Variable (Previous Chapter) (This Chapter)

6. 6 The Normal Distribution

7. 7 ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median X f(X) μ σ

8. 8 The Normal Distribution You can obtain probabilities for the normal distribution using table 2 You can look up Z (the number of standard deviations above or below the mean) on the left and top of this table and then the numbers inside the table will give you the probabilities to the LEFT of Z Z

9. 9 The Normal Distribution The following formula is useful for finding Z (the number of standard deviations above or below the mean) Z itself has what is called a standard normal distribution (mean=zero and standard deviation=1)

10. 10 In class exercise #83: Exam scores have a normal distribution with a mean of 70 and a standard deviation of 10. Find the percentage of students who score less than 85.

11. 11 In class exercise #84: Exam scores have a normal distribution with a mean of 70 and a standard deviation of 10. a) Find the percentage of students who score less than 45 b) Find the percentage of students who score greater than 45 c) Find the percentage of students who score between 52 and 90 d) Find the percentage of students who score greater than 80 or less than 60. e) Find the percentage of students who score less than 5

12. 12 The Normal Distribution – Working Backwards Sometimes you have the probability and want to know the value of the normal random variable that corresponds to that probability For example, if I want to give 10% of students a failing grade on the exam, what should be the cutoff? In these case, the following formula is useful

13. 13 In class exercise #85: Exam scores have a normal distribution with a mean of 70 and a standard deviation of 10. a) Find the cutoff that will give 10% an F. b) Find the cutoff that will give 20% an A. c) Find the cutoffs that will give the middle 40% a C.

14. 14 Sampling Distributions A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population We will study the sampling distribution of the sample mean

15. 15 In class exercise #86: The data at http://www.cob.sjsu.edu/mease_d/newfreethrows.xls has updated free throw percentages for all NBA players. 1) Use Excel to compute the (population) mean and variance. 2) Draw some simple random samples of size n=4 and compute the sample means for each of these. 3) Compute the mean of these sample means. How does that compare that to the population mean? 4) Compute the variance of these sample means. How does that compare that to the population variance?

16. 16 The Sampling Distribution of the Sample Mean

17. 17 The Sampling Distribution of the Sample Mean If the original population has mean and standard deviation then the sampling distribution of the sample mean will have a mean of and a standard deviation of. Further, it will follow a normal distribution if 1) the original population has a normal distribution OR 2) the sample size n is large enough (bigger than 30) (this is called the Central Limit Theorem)

18. 18 In class exercise #87: (This is question 44 part A for your homework) 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, what is the probability the mean will be between 7.8 and 8.2 minutes?