Normal Distribution
Z scores
100

A ___ is the number of standard deviations that a data value is from the mean?

Z-score

100

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

200

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

200
A test has a mean score of 80 and a standard deviation of 10.  What z score corresponds to a test score of 65?

z = (65-80)/10 = -1.5

300

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

300

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

400

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

400

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%

500

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

500

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