CSC 323 Quarter Winter 0203 Daniela Stan Raicu

CSC 323 Quarter: Winter 02/03 Daniela Stan Raicu School of CTI, De. Paul University 2/1/2022 Daniela Stan - CSC 323 1

Outline Ø Standard Normal Distribution Ø Introduction to the Statistical Software SAS 2/1/2022 Daniela Stan - CSC 323 2

Normal Distributions • The heights of adult women in the United States follow, at least approximately, a bell-shaped curve. What do you think that means? The most adult women are clumped around the average, with numbers decreasing the farther values are from the average in either direction. • Let us assume that the average of the heights of adult women is = 65 and the standard deviation is = 2. 5. What does the 68 -95 -97. 7 rule imply? 2/1/2022 Daniela Stan - CSC 323 3

65 -2. 5 65 65+2. 5 • 68% of adult women have heights between 62. 5 and 67. 5 inches; 95% of adult women have heights between 60 and 70 inches; 99. 7% have heights between 67. 5 and 72. 5 inches 2/1/2022 Daniela Stan - CSC 323 4

Standardized score A “standardized score” is simply the number of standard deviations an individual falls above or below the mean for the whole group. • Values above the mean have positive standardized scores; values below the mean have negative ones. Example: Females (ages 18 -24) have a mean height of 65 inches and a standard deviation of 2. 5 inches. What is the standardized score of a a women who is 67. 5 inches tall? Standardized score: = (67. 5 – 65)/2. 5=1 2/1/2022 Daniela Stan - CSC 323 5

Standardized Scores standardized score = (observed value - mean) / (std dev) · · z is the standardized score x is the observed value is the population mean is the population standard deviation 2/1/2022 Daniela Stan - CSC 323 6

The standard normal distribution Ø The standard normal distribution N(0, 1) is the normal distribution with mean 0 and standard deviation 1 Ø If a variable X has any normal distribution N( , ), then the standardized variable Z has the standard normal distribution N(0, 1). Ø Why are normal distributions so important? Ø They are good descriptions for real data Ø Many statistical inference procedures based on normal distributions work well for other roughly symmetric distributions. 2/1/2022 Daniela Stan - CSC 323 7

Normal distribution calculations Example: The heights of young women are approximately normal with mean =64. 5 inches and =2. 5 inches. What is the proportion of women how are less than 68 inches tall? 1. State the problem: X = height, X < 68 2. Standardize: 68 standardized to 1. 4 X<68 2/1/2022 Daniela Stan - CSC 323 Z < 1. 4 8

Normal distribution calculations 3. What proportion of observations/women on the standard normal variable Z take values less than 1. 4? Table entry is area to the left of z Area (Z<1. 4)=. 9192 Table A at the end of the book gives areas (proportions of observations) under standard normal curve. 2/1/2022 Daniela Stan - CSC 323 9