The Normal Distribution

Chapter 2Part 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

50.5 55.8 61.1 66.4

Iliana

Lanier’s and McCallum’s distributions

50.5 55.8 61.1 66.4

Iliana

77.2 85.3 93.4 101.5

Iliana

•How can we find Iliana’s relative position?

Iliana’s score – class average standard deviation

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: z 44 328 15.

Second quiz: z 28 233

1667.

Relative to their departments, which is better paid if the husband earns $14.60 and the wife earns $15.75?

z

14 60 12 80120

15. ..

.Husband:

Wife: z

15 75 1350180

125. ..

.

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.

z 15.93nd percentile

What percentile is the husband located in his department?

z 125. P z( . ) . . 125 1 8944 1056

What percent of employees in the wife’s department earn better than her?

What would the wife need to earn to match her husband’s relative position?

z

14 60 12 80120

15. ..

.Husband:

Wife: z

1575 1350180

125. ..

.

x

1350180

15..

. 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!

z

14 60 12 80120

15. ..

.

x 12 80120

.. 165.

x 12 80 198. .x $14.78

The husband will need to earn at least $14.78 to be in the 95th percentile.

2

– 2

14.24 .0228P x 59.60 .9772P x

14.24 59.60 .9772 .0228 .9544P x

89%

44.5%

5.5%

z-score = –1.60 z-score = 1.60

94.5%

36.921.6011.32x

36.921.60

11.32x

The middle 89% of the data ranges from 18.81 to 55.03 ppb.