1 Business 90: Business Statistics Professor David Mease Sec 03, T R 7:30-8:45AM BBC 204 Lecture 19 = More of Chapter “The Normal Distribution and Other Continuous Distributions” (TNDAOCD) Agenda: 1) Go over quiz over Homework 6 2) Reminder about Homework 7 (due Thursday 4/22) 3) Lecture over more of Chapter TNDAOCD

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

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 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 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

7 The Normal Distribution

8 ‘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

9 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

10 The formula for the normal probability density function is Where e = the mathematical constant approximated by π = the mathematical constant approximated by μ = the population mean σ = the population standard deviation X = any value of the continuous variable The Normal Distribution

11 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

12 In class exercise #75: Use table 2 to determine the probability that a normally distributed random variable is less than 1 standard deviation above the mean.

13 In class exercise #76: Use table 2 to determine the probability that a normally distributed random variable is greater than 1.6 standard deviations above the mean.

14 In class exercise #77: Use table 2 to determine the probability that a normally distributed random variable is greater than 1.63 standard deviations above the mean.

15 In class exercise #78: Use table 2 to determine the probability that a normally distributed random variable is greater than 2.34 standard deviations below the mean.

16 In class exercise #79: Use table 2 to determine the probability that a normally distributed random variable is less than two standard deviations above the mean but greater than two standard deviations below the mean. Where have you seen this probability before?

17 In class exercise #80: 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.

18 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)

19 In class exercise #81: 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

20 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

21 In class exercise #82: 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.