Normal distribution Flashcards
Module 9: Handout 19, 20
Normal distribution is a
continuous probability distribution
The normal distribution z scores are described by the
68-95-99.7 rule
What is a density function?
A graph that is measured by the area underneath the line created, similar to a bar graph, but curved
To find a z-score of a normal distribution
(x minus the “mean/Myu”) divided by the standard deviation
What is the purpose of a z-score
Let’s us calculate the probability that that score would come up in our normal distribution
Steps to finding the probability of a certain question
1) Find z-score in terms of the x value given, to find the z-score you need the X value, the mean, and the standard deviation
2) Use z-table and find the z-score number
3) Convert the decimal number to the percentage
4) That percentage is the probability
Example 1 What is the probability that a box will be underweight (Under 16 oz.) If boxes are being filled to 16.4 oz.
1) Find mean, X, and SD
2) Mean=16.4 X=16 SD=0.3
3) Find Z-score (16-16.4)/(0.3)
4) Z-score=-1.33
5) Use z-table to find percentage of -1.33
6) On z-table, find -1.3, then the 0.03
7) The number given will be 0.09176=0.0918
8) Turn it into percentage
9) 0.0918=9.18%=9.2%
10) 9.2% chance that box will be underweight
Example 1 (on calculator) What is the probability that a box will be underweight?
1) “2nd”, “vars”, “2”
2) Normalcdf(x1, x2, Myu/mean, standard deviation)
3) For x1, the input is “negative infinity sign” so write -E99
4) -E99 is “2nd”, “comma key”, “99”
5) x2 is the X value which equals 16 since we’re trying to find the percentage between 16 and under
6) The mean is 16.4
7) Standard deviation is 0.3
8) “Normalcdf(-E99, 16, 16.4, 0.3)”
9) Answer=9.1%
It is a slightly different answer since it doesn’t round up
Example 1 (on calculator) What is the probability that a box will be overweight if overweight = 17oz. and the mean is 16.4oz
1) “2nd”, “vars”, “2”
2) Normalcdf(x1, x2, Myu/mean, standard deviation)
3) Normalcdf(17, E99, 16.4, 0.3)
4) Answer = 0.028 or 2.3%
Example 1 What is the probability that the box will weigh between 15.9 and 16.9 ounces?
1) Find both z-scores
2) ((15.9 - 16.4)/0.3) = -1.67 and ((16.9 - 16.4)/0.3) = 1.67
3) Use Z-table to find areas
4) Using z-table numbers, subtract the answers from each other
5) 0.9525 - 0.0475 = 0.905
6) The area between 15.9 and 16.9 is 0.905
Example 2 How to find percentile values such as “P3” which separates 3% and 97% of the distribution
1) We find x in equation z=(x-myu)/SD
2) So, use x=(SD times z)/myu
3) So on Z-table, find 3% or specifically 0.0301 or close to 0.03
4) Find the z value which are the numbers that form the number 0.0301
5) So it is -1.8 and 0.08
6) z = 1.8+0.8 = 1.88 which is -1.88
7) Then input variables into equation
8) x=(SD)(z)+Myu=(0.3)(-1.88)+16.4=15.836
9) So P3 or 3% weigh less than 15.8
Example 2 Calculator How to find percentile values such as “P3” which separates 3% and 97% of the distribution
1) “2nd”“vars”“3”“invnorm(percentage, mean, SD)”
2) invnorm(0.03, 16.4, 0.3)
3) invnorm(0.03, 16.4, 0.3)=15.8
Example 3 A light bulb has a mean time of 245 hours before it burns out, with a
standard deviation of 40 hours. Find P80
1) Use x = (SD)(z)+Myu
2) Find SD = 40
3) Find z = 0.8+0.04 = 0.84
btw found on Z-table
4) Myu or mean = 245
5) (40)(0.84)+245=278.6
6) So 80%
Example 3 Calculator A light bulb has a mean time of 245 hours before it burns out, with a
standard deviation of 40 hours. Find P80
1) Invnorm(0.8, 245, 40)=278.7
2) 0.8 is 80%, 245 is the mean, 40 is the standard deviation
Example 3 what is x and the point of SD?
X is the number of hours it takes until it burns out. Given a percentage such as P80 or 80%, we find the number of hours it takes until 80% of the lightbulbs burns out. So, 80% of lightbulbs last for 278.6 hours
