5/18/10 Lecture 6 1STOR 155 Introductory StatisticsLecture 6: The Normal Distributions (II)The UNIVERSITY of NORTH CAROLINAat CHAPEL HILL5/18/10 Lecture 6 2Review• Density curves• Normal distributions and normal curves• The 68-95-99.7 rule for normal distributions• Standardizing observations• The standard normal distribution5/18/10 Lecture 6 3Topics• The standard normal table • Normal distribution calculation• Normal quantile plot5/18/10 Lecture 6 4The 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?Answer: N(a, |b|) (Be careful with the sign of b,more later.)5/18/10 Lecture 6 5Table 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.5/18/10 Lecture 6 6Table 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.5/18/10 Lecture 6 75/18/10 Lecture 6 8Example• If Z has a standard normal distribution, determine the value z for which the area under the normal curve between 0 and z is0.4192.• z=1.4 or -1.45/18/10 Lecture 6 9Example• zAis defined as the z value for which the area to the right of zAunder the standard normal curve is A. • Determine z0.0808.• 1.45/18/10 Lecture 6 10Example: 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?5/18/10 Lecture 6 11Normal distribution• If a variable X has a normal distribution with mean and standard deviation , denoted by N( , ), then the standardized variablehas 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.XZx5/18/10 Lecture 6 12Example• The heights of young women follow N(64.5, 2.5). What is the proportion of young women who are shorter than 66 inches?5/18/10 Lecture 6 13Solution1. 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 Z5/18/10 Lecture 6 14A letter to AbbyDear 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 Reader5/18/10 Lecture 6 15A 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)?5/18/10 Lecture 6 16Example 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?5/18/10 Lecture 6 175/18/10 Lecture 6 18The Normal Quantile Plot• Normal distributions: nice models for a lot of data.• A lot of nice calculation can be done if assuming normality.• Normality does not always hold!!!– 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.5/18/10 Lecture 6 19How 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.5/18/10 Lecture 6 20Use 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.5/18/10 Lecture 6 21Histograms of 3 Variables0 1 2 3 4 5 6-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 35/18/10 Lecture 6 22Normal Quantile Plots of the 3 Variables (skewed or with fat-tails ?)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 Plot5/18/10 Lecture 6 23-3 -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 Plot5/18/10 Lecture 6 24Speed of Light5/18/10 Lecture 6 25Speed of Light (no outliers)5/18/10 Lecture 6 26IQ scores of 7-graders5/18/10 Lecture 6 27Take Home Message• The standard normal table• Normal distribution calculation• Normal quantile
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