STAT 155 Introductory Statistics. Lecture 6: The Normal Distributions (II)

Transcription

1 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 6: The Normal Distributions (II) 9/14/06 Lecture 6 1

2 Review Density curves Normal distributions and normal curves The rule for normal distributions Standardizing observations The standard normal distribution 9/14/06 Lecture 6 2

3 Topics The standard normal table Normal distribution calculation Normal quantile plot 9/14/06 Lecture 6 3

4 The standard normal distribution The standard normal distribution is the normal dist. with mean 0 and standard deviation 1, denoted as N(0,1). N(0,1) can be treated as a benchmark. Any normal distribution can be related to N(0,1) by a linear transformation. Z: N(0,1) What is the distribution for X=a+bZ? 9/14/06 Lecture 6 4

5 Table A: The Standard Normal Table Table A is a table of areas under the standard normal density curve. The table entry for each value z is the area under the curve to the left of z. 9/14/06 Lecture 6 5

6 Table A : The Standard Normal Table Table A can be used to find the proportion of observations of a variable which fall to the left of a specific value z if the variable follows a normal distribution. 9/14/06 Lecture 6 6

7 9/14/06 Lecture 6 7

8 Example If Z has a standard normal distribution, determine the value z for which the area under the normal curve between 0 and z is z=1.4 or /14/06 Lecture 6 8

9 Example z A is defined as the z value for which the area to the right of z A under the standard normal curve is A. Determine z /14/06 Lecture 6 9

10 Example: Young Women s Height The z-scores of young women s heights are approximately standard normal. % of z-scores between -1 and 1? % of z-scores lower than -1 or higher than 2? % higher than 1.4? 9/14/06 Lecture 6 10

11 Normal distribution If a variable X has a normal distribution with mean and standard deviation σ, denoted by N( µ, σ ), then the standardized variable µ = X σ has the standard normal distribution. Z µ The area to the left of x under the density curve for X is the same as the area to the left of x µ under the density curve for Z. σ Table A can be used for any normal distribution Bridge: standardizing and z-score. 9/14/06 Lecture 6 11

12 Example The heights of young women follow N(64.5,2.5). What is the proportion of young women who are shorter than 66 inches? 9/14/06 Lecture 6 12

13 Solution 1. State the problem: Let X denote the height of a randomly chosen young woman, then X follows N(64.5,2.5). We want the proportion of young women with X< 66 inches. 2. Standardize: Transform X to a standard normal variable Z. X <66 (X )/2.5 < ( )/2.5 Z < Use the table: From Table A, we find that the proportion of young women with height < 66 inches is About 73 % of young women is shorter than 66 inches. 9/14/06 Lecture 6 13

14 A letter to Abby Dear Abby: You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby for 10 months and 5 days, and there is no doubt about it because I know the exact date my baby was conceived. My husband is in the Navy and it couldn't have possibly been conceived any other time because I saw him only once for an hour, and I didn't see him again until the day before the baby was born. I don't drink or run around, and there is no way the baby isn't his, so please print a retraction about the 266-day carrying time because I am in a lot of trouble. - San Diego Reader 9/14/06 Lecture 6 14

15 A letter to Abby According to well-documented norms, the distribution of gestation time is approximately normal with mean 266 days and SD 16 days. What percent of babies have a gestation time greater or equal to 310 days (10 months and 5 days)? 9/14/06 Lecture 6 15

16 Example 1.30: Inverse problem Scores on the SAT verbal test in recent years follow approximately the N(505, 110) distribution. How high must a student score in order to be placed in the top 10% of all students taking the SAT? 9/14/06 Lecture 6 16

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18 The Normal Quantile Plot Normal distributions: nice models for a lot of data. A lot of nice calculation can be done if assuming normality. Normality is not everywhere!!! Economic variables: personal income, gross sales of business Financial variables: stock/option price Other variables: conversation time Dangerous to assume normality without actually testing it. The normal quantile plot is a graphical tool, which can be used to decide whether the data come from a normal distribution. 9/14/06 Lecture 6 18

19 Histograms of 3 Variables /14/06 Lecture 6 19

20 How does a normal quantile plot work? Sort the obs from smallest to largest; Record what percentile of the data each ob occupies; Do normal distribution calculations to find the z- scores at the same percentiles; Plot each data point x against the corresponding z. If the data are close to normal, then the points will lie close to some straight line. 9/14/06 Lecture 6 20

21 Use of Normal Quantile Plots If the points on a normal quantile plot lie close to a straight line, the plot indicates the data are normal. Systematic deviations from a straight line indicate a non-normal distribution. Outliers appear as points that are far away from the overall pattern of the plot. 9/14/06 Lecture 6 21

22 Histograms of 3 Variables /14/06 Lecture 6 22

23 Normal Quantile Plots of the 3 Variables Normal Quantile Plot Normal Quantile Plot Normal Quantile Plot 9/14/06 Lecture 6 23

24 Normal Quantile Plot Normal Quantile Plot 9/14/06 Lecture 6 24

25 Speed of Light 9/14/06 Lecture 6 25

26 Speed of Light (no outliers) 9/14/06 Lecture 6 26

27 IQ scores of 7-graders 9/14/06 Lecture 6 27

28 Take Home Message The standard normal table Normal distribution calculation Normal quantile plot 9/14/06 Lecture 6 28