Normal Distribution
A ___ is the number of standard deviations that a data value is from the mean?
Z-score
The mean weight of 5 year old boys in a certain area is 69 pounds with a standard deviation of 2.5 inches. What is the z-score for a boy who is 72 pounds?
z = (72-69)/(2.5) = 1.2
You took the SAT and scored 600. Your best friend took the ACT and scored 21. If the SAT has a mean score of 500 with a standard deviation of 100 and the ACT has a mean score of 15 with a standard deviation of 4, which score is better yours or your best friend’s?
Your best friends
1. (600-500)/100 = 1
2. (21-15)/4 = 1.5
z = (65-80)/10 = -1.5
An IQ tests’ results are normally distributed with a mean score of 90 and a standard deviation of 14. In what interval will the middle 95% of the IQ scores be?
x low = 90 - 1.96 x 14 = 90 -27.44 = 62.6
x high = 90 + 1.96 x 14 = 90 + 27.44 = 117.44
between 62.6 and 117.4
The average lifter at a local gym can bench press 200 pounds with a standard deviation of 25 pounds. What weight corresponds to a Z score of 1.5?
200 + (1.5x2.5) =3.75 200 + 37.5 = 237.5
SAT scores are nearly normal with a mean score of 600 and a standard deviation of 100. What proportion of test takers score over 400?
97.7%
Z = -2
M = 600
Sd = 100
Normalcdf(-2, 1e99, 0, 1) = .977
In a certain town, the average annual average rainfall is 32 inches with a standard deviation of 4 inches. What is the probability that in a given year there will be rainfall less than 27 inches?
z= (27-32)/4 =-1.25 -1.25
normalcdf(-1e99, -1.25, 0, 1) = 10.56%
SAT scores are nearly normal with a mean score of 500 and a standard deviation of 100. The bottom 40% of students must have scored below what SAT score?
474.64
A company wants to award bonuses To employees whose productivity is in the top 15% of the distribution. If productivity is normally distributed with a mean of 60 and a standard deviation of 8, what is the minimum productivity score required to receive a bonus?
top 15% according to empirical rule = 1.04
60 + (1.04)(8) = 68.32