The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 6 The Normal Distributions II 9 15 09 Lecture 6 1 Review Density curves Normal distributions and normal curves The 68 95 99 7 rule for normal distributions Standardizing observations The standard normal distribution 9 15 09 Lecture 6 2 Topics The standard normal table Normal distribution calculation Normal quantile plot 9 15 09 Lecture 6 3 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 15 09 Lecture 6 4 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 15 09 Lecture 6 5 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 15 09 Lecture 6 6 9 15 09 Lecture 6 7 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 4 9 15 09 Lecture 6 8 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 4 9 15 09 Lecture 6 9 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 15 09 Lecture 6 10 Normal distribution If a variable X has a normal distribution with mean and standard deviation denoted by N then the standardized variable Z X 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 x under the density curve for Z Table A can be used for any normal distribution Bridge standardizing and z score 9 15 09 Lecture 6 11 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 15 09 Lecture 6 12 Solution 1 2 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 Standardize Transform X to a standard normal variable Z X 66 X 64 5 2 5 66 64 5 2 5 Z 0 6 3 9 15 09 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 Lecture 6 13 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 15 09 Lecture 6 14 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 15 09 Lecture 6 15 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 15 09 Lecture 6 16 9 15 09 Lecture 6 17 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 15 09 Lecture 6 18 Histograms of 3 Variables 3 0 1 2 3 4 5 3 9 15 09 2 1 0 1 2 6 2 1 0 Lecture 6 1 2 3 19 3 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 zscores 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 15 09 Lecture 6 20 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 15 09 Lecture 6 21 Histograms of 3 Variables 3 0 1 2 3 4 5 3 9 15 09 2 1 0 1 2 6 2 1 0 Lecture 6 1 2 3 22 3 Normal Quantile Plots of the 3 Variables 20 001 01 05 10 25 50 75 90 95 99 999 001 01 05 10 25 50 75 90 95 99 999 10 20 0 10 10 20 3 0 001 01 05 10 25 50 75 90 95 99 999 30 3 2 1 0 1 2 3 4 2 3 Normal Quantile Plot 1 2 1 0 1 2 3 Normal Quantile Plot 0 1 2 3 3 2 1 0 1 2 3 4 Normal Quantile Plot 9 15 09 Lecture 6 23 4 20 001 01 05 10 25 50 75 90 95 99 999 10 0 10 20 30 3 3 2 1 0 1 2 3 2 1 0 1 2 3 4 Normal Quantile Plot 3 001 01 05 10 25 50 75 90 95 …