Standardizing NPD The Standard Normal Probability Distribution is
Standardizing NPD
The Standard Normal Probability Distribution … is a normal distribution with a mean of 0 and a standard deviation of 1. Also called the Z Distribution A z-value is the distance between a selected value (designated X) and the population mean, divided by the population A Standard Deviation, . z-value is, therefore, a location on the standard normal curve. Formula
Finding Probabilities for Other Normal Curves • To find the probabilities for other normal curves, standardize the relevant values and then use the table of z areas. • If x is a random variable whose behavior is described by a normal distribution with mean m and standard deviation s , then P(x < b) = P(z < b*) P(x > a) = P(z > a*) Where z is a variable whose distribution is standard normal and
Example-1: Data on the length of time to complete registration for classes using an on-line registration system suggest that the distribution of the variable x = time to register for students at a particular university can well be approximated by a normal distribution with mean m = 12 minutes and standard deviation s = 2 minutes.
x = time to register m = 12 minutes and s = 2 Standardized minutes this value. a) What is the probability that it will take a randomly selected student less to Look thisthan value 9 upminutes in the table. complete registration? P(x < 9) =. 0668 9
x = time to register this value. m = 12 minutes and s = 2 Standardized minutes b) What is the probability that it will take a randomly selected student. Look more thisthan value 13 upminutes in the to complete registration? table and subtract from 1. P(x > 13) = 1 -. 6915 =. 3085 13
x = time to register m = 12 minutes and s = 2 minutes c) What is the probability that it will take a randomly selected student between 7 and 15 minutes to complete registration? P(7 < x < 15) =. 9332 -. 0062 =. 9270 7 15
Use GDC to solve the following: What is the probability that it will take a randomly selected student between 7 and 15 minutes to complete registration? x = time to register m = 12 minutes and s = 2 minutes 7 15
Example-2 A Company produces “ 20 ounce” jars of a picante sauce. The true amounts of sauce in the jars of this brand sauce follow a normal distribution. Suppose the companies “ 20 ounce” jars follow a N(20. 2, 0. 125) distribution curve. (i. e. , The contents of the jars are normally distributed with a true mean = 20. 2 ounces with a true standard deviation = 0. 125 ounces. a) What proportion of the jars are under-filled? b) What proportion of the sauce jars contain between 20 and 20. 3 ounces of sauce? c) 99% of the jars of this brand of picante sauce will contain more than what amount of sauce?
x = time to register m = 12 minutes and s = 2 minutes d) Because some students do not log off properly, the Use the formula for standardizing to find x. university would like to log off students Look up the area to the left automatically after some timeof has It is decided to select this a*elapsed. in the table. time so that only 1% of students will be automatically logged off while still trying to register. What time should the automatic log off be set at? P(x > a*) =. 01 a* = 16. 66 . 99 . 01 a*
There are two places to see normal table: 1. In Introduction to Statistics Course 2. I can’t think of any other place
What to show to receive full mark? Name Parameters Boundary Direction or drawing graph is the safest method. “Just draw the damn picture!” Mean, s. d, lower bound and upper bound Normalcdf(11, 14, 5, 1. 5) doesn’t give you full credit • •
Mr. Webber has determined that the scores in his Literature course are approximately normally distributed with a mean of 87 and a standard deviation of 4. He announces to the class that the top 15% of the scores will earn an A. What is the lowest score a student can earn and still receive an A?
Pair-work
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