University:
California State University, Northridge
Course:
MATH 140 | Introductory Statistics
Academic year:

2009
Views:

109

Pages:

11
Author:

sammerito22923g0

Math 140 Introductory Statistics Lecture 7 The Two Main Problems in the Standard Normal Distribution. Unknown Percentage. (Given z, find P ) With Table A (end of the textbook) Use the units and the first decimal to locate the row and the closest hundredths digits to locate the column. The number found is the percentage of the number of values below z. With Calculator Enter normalcdf(-99999, z) to get the percentage of the number of values below z. Example (given z find P) Calculator P = normalcdf(-99999,1.23) = .8906513833 ~ 89.07% Table A Look for -row labeled 1.2 -column labeled .03 The intersection shows P =.8907 = 89.07% The Two Main Problems in the Standard Normal Distribution. Unknown Value Problem. (Given P, find z ) With Table A Look for P in the body of the table. (or the number closest to it). Read back the row and column for that number. Use the row as the units and tenths of z, and the column as the hundredths digits of z. Note that P must be a percentage (written as a proportion, that is, a number between 0 and 1) of the number of values below a certain value z. With Calculator Enter invNorm(P) to get the value z such that P equals the percentage of the number of values below z. Example (given P find z-score) Calculator z=invNorm(.75) = .6744897495 ~ .67 Table A The value closest to .75 in the body of table A is .7486, which is in row .6 and column .07. Then the z-score is .67 Value ↔ z-score Standardizing (from (x ↔ z ) x to z) x−x z= SD Unstandardizing (from z to x) x = x + z (SD) The two main problems (summary) Unknown percentage given x, find P x to z to P Unknown value given P, find x P to z to x x−x z= SD invNorm(P) normalcdf(-99999, z) x = x + z (SD) Table: row and column Table: body Example (p. 88 – given x find P) Standardize (get z) x = 74 x − x 74 − 70.1 z= = = 1.4444 SD 2.7 Percentage below 74 in P = normalcdf(-99999,1.4444) ~ .9257 = 92.57% Percentage above 74 in 1-.9257 = .0743 = 7.43% Example (p. 89 – given P find x) Get z-score P = 75% = .75 (given) z =invNorm(.75) = .6744897495 ~ .67 Unstandardize (get x) x = x + z ( SD ) = 64.8 + .67(2.5) = 66.475 in Example (p. 91 – given P find x) According to the table on page 87, the distribution of death rates from cancer per 100,000 residents by state is approximately normal*, with mean 196 and SD 31. The middle 90% of death rates are between what two numbers? *Provided that Alaska and Utah, which are outliers because of their unusually young populations, are left out. Example (p. 91 – given P find x) cont. According to the table on page 87, the distribution of death rates from cancer per 100,000 residents by state is approximately normal*, with mean 196 and SD 31. The middle 90% of death rates are between what two numbers? Get z-scores (middle 90% is betwee 5% and 95%) 5% =.05 corresponds to z = -1.64485 95%= .95 corresponds to z = 1.64485 Unstandardize x = x + z ( SD) = mean + z ( SD) = 196 + (−1.64485)(31) = 145.00965 x = x + z ( SD) = mean + z ( SD) = 196 + (1.64485)(31) = 246.99035 So the middle 90% of states have death rates between 145.009 and 246.99 deaths per 1000,000 residents. *Provided that Alaska and Utah, which are outliers because of their unusually young populations, are left out.