Transcript Class Notes - Mr. Taylor`s Math

The Normal Distribution
Chapter 2
Part 2
Iliana’s Grade
• After 5 weeks of class Iliana must transfer
from a stat class at Lanier to this class. Last
week was the chapter 1 test in both classes.
Iliana scored a 61 out of 70. Let’s say our
test was out of 100 points. What score
should she be given?
Iliana’s claim
• Iliana claims that her test at Lanier was
harder than our test.
• Does your previous method of assigning a
grade take in consideration difficulty?
• If we have all of the data, what important
facts can we utilize to improve our
assignment of Iliana’s grade?
Important Facts
• Maximum possible on our test was 100 pts
while Lanier’s test was 70 pts.
• Mean score on Lanier’s test was 50.5 pts
while our test was 77.2 pts.
• Standard deviation on Lanier’s test was
5.3 pts while ours was 8.1 pts.
• Test scores from both high schools tend to
be normally distributed.
• How will we fairly assign Iliana’s score?
Lanier’s distribution
Iliana
50.5 55.8 61.1 66.4
Lanier’s and McCallum’s distributions
Iliana’s score – class average
standard deviation
Iliana
Iliana
50.5 55.8 61.1 66.4
77.2 85.3 93.4 101.5
•How can we find Iliana’s relative position?
Formula
• What is the formula to find the relative
position for any distribution?
Iliana’s score – class average
standard deviation
z–score=
x

1. Suppose as student has taken two quizzes in a statistics course. On
the first quiz the mean score was 32, the standard deviation was 8, and
the student received a 44. The student obtained a 28 on the second
quiz, for which the mean was 23 and the standard deviation was 3. If
test scores are approximately normal, on which quiz did the student
perform better relative to the rest of the class?
z
x

First quiz:
Second quiz:
44  32
z
 15
.
8
28  23
 1.667
z
3
3. A married couple is employed by the same company. The husband
works in a department for which the mean hourly rate is $12.80 and the
standard deviation is $1.20. His wife is employed in a department where
the mean rate is $13.50 and the standard deviation is $1.80.
Relative to their departments, which is better paid if the husband earns
$14.60 and the wife earns $15.75?
Husband:
Wife:
14.60  12.80
z
 15
.
120
.
15.75  1350
.
z
 125
.
180
.
What percentile is the husband located in his department?
z  15
.
 93
nd
percentile
What percent of employees in the wife’s department earn
better than her?
z  125
.
P( z  125
. )  1.8944 .1056
What would the wife need to earn to match her husband’s
relative position?
Husband:
Wife:
14.60  12.80
z
 15
.
120
.
15.75  1350
.
z
 125
.
180
.
x  1350
.
 15
.
180
.
x  1350
.  2.7
x  $16.20
The wife would need to earn $16.20 to match the husband’s relative
position.
If the husband wanted to earn in the 95th percentile, how
much should he earn per hour?
Need a z-score of 1.65!
14.60  12.80
z
 15
.
120
.
x  12.80
 165
.
1.20
x  12.80  198
.
x  $14.78
The husband will need to earn at least $14.78 to be in the 95th percentile.
–2
P  x  14.24   .0228
2
P  x  59.60   .9772
P 14.24  x  59.60   .9772  .0228  .9544
5.5%
94.5%
89%
44.5%
z-score = –1.60
x  36.92
1.60 
11.32
z-score = 1.60
x  36.92
1.60 
11.32
The middle 89% of the data ranges from 18.81 to 55.03 ppb.