2/1/11 Lecture 6 1 STOR 155 Introductory Statistics Lecture 6: The Normal Distributions (II) The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL2/1/11 Lecture 6 2 Review • Density curves • Normal distributions and normal curves • The 68-95-99.7 rule for normal distributions • Standardizing observations • The standard normal distribution2/1/11 Lecture 6 3 Topics • The standard normal table • Normal distribution calculation • Normal quantile plot2/1/11 Lecture 6 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?2/1/11 Lecture 6 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.2/1/11 Lecture 6 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.2/1/11 Lecture 6 72/1/11 Lecture 6 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 0.4192. • z=1.4 or -1.42/1/11 Lecture 6 9 Example • zA is defined as the z value for which the area to the right of zA under the standard normal curve is A. • Determine z0.0808. • 1.42/1/11 Lecture 6 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?2/1/11 Lecture 6 11 Normal distribution • If a variable X has a normal distribution with mean and standard deviation , denoted by N( , ), then the standardized variable has the standard normal distribution. • The area to the left of x under the density curve for X is the same as the area to the left of under the density curve for Z . • Table A can be used for any normal distribution • Bridge: standardizing and z-score. XZx2/1/11 Lecture 6 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?2/1/11 Lecture 6 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. 3. Use the table: From Table A, we find that the proportion of young women with height < 66 inches is 0.7257. About 73 % of young women is shorter than 66 inches. 66 X 64.5)/2.5-(66 64.5)/2.5-(X 0.6 Z2/1/11 Lecture 6 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 Reader2/1/11 Lecture 6 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)?2/1/11 Lecture 6 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?2/1/11 Lecture 6 172/1/11 Lecture 6 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.2/1/11 Lecture 6 19 Histograms of 3 Variables 0 1 2 3 4 5 6-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 32/1/11 Lecture 6 20 How does a normal quantile plot work? • Sort the observations from smallest to largest; • Record what percentile of the data each obs. 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. • See the two files: Normal Quantile Plots in Excel.doc and NormQuant.xls2/1/11 Lecture 6 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.2/1/11 Lecture 6 22 Histograms of 3 Variables 0 1 2 3 4 5 6-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 32/1/11 Lecture 6 23 Normal Quantile Plots of the 3 Variables 01020.001 .01 .05.10 .25 .50 .75 .90.95 .99 .999-3 -2 -1 0 1 2 3 4Normal Quantile Plot-30-20-1001020.001 .01 .05.10 .25 .50 .75 .90.95 .99 .999-3 -2 -1 0 1 2 3 4Normal Quantile Plot-3-2-10123.001 .01 .05.10 .25 .50 .75 .90.95 .99 .999-3 -2 -1 0 1 2 3 4Normal Quantile Plot2/1/11 Lecture 6 24 -3 -2 -1 0 1 2 3-30-20-1001020.001 .01 .05.10 .25 .50 .75 .90.95 .99 .999-3 -2 -1 0 1 2 3 4Normal Quantile Plot-3 -2 -1 0 1 2 3-3-2-10123.001 .01 .05.10 .25 .50 .75 .90.95 .99 .999-3 -2 -1 0 1 2 3 4Normal Quantile Plot2/1/11 Lecture 6 25 Speed of Light2/1/11 Lecture 6 26 Speed of Light (no outliers)2/1/11 Lecture 6 27 IQ scores of 7-graders2/1/11 Lecture 6 28 Take Home Message • The standard normal table • Normal
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