1. 3.5 Applying the Normal Distribution: Z-Scores
How can the Normal Distribution be used to accurately determine the percentage of data that lies above or below a given data value?
How can the Normal Distribution be used to compare data from two different data sets?
Enter…
“The Standard Normal Distribution”

2. 3.5 Applying the Normal Distribution: Z-Scores
Two students from different schools have a mark in a MDM 4U class.
Adam has a mark of 83
Brenda has a mark of 84
Who has a better grade?

3. 3.5 Applying the Normal Distribution: Z-Scores
Not so easy to answer…
Adam’s class average is 70 with a sdev of 9.8
Brenda’s class average is 74 with a sdev of 8
(Assume a Normal Distribution of marks in both classes)
To make the comparison – must STANDARDIZE the distributions.

4. 3.5 Applying the Normal Distribution:Z-Scores
Standard Normal Distribution
Special normal distribution with a mean of 0 and a standard deviation of 1
X~N(0,12)
N(2.1, 9) ≡ mean of 2.1 and sdev of 3
Each element of a normal distribution can be translated to the same place on a Standard Normal Distribution by determining the number of standard deviations a given score lies away from the mean.

5. 3.5 Applying the Normal Distribution:Z-Scores
For a given score, x, we can say:
‘z’ is the number of sdev the score lies above or below the mean
Solving for z:
This is the z-score of a piece of data

6. 3.5 Applying the Normal Distribution: Z-Scores
A positive z-score indicates the value lies above the mean
A negative z-score indicates the value lies below the mean

7. 3.5 Applying the Normal Distribution: Z-Scores
Calculating the z-scores for Adam & Brenda
Now you can see that Adam has the better mark as he is 1.33 sdevs from the mean vs 1.25 sdevs in Brenda’s case.

8. 3.5 Applying the Normal Distribution: Z-Scores
Z-Score Table
Used to find the proportion of data that has an equal or lesser score than a given value under the standardized normal distribution curve
Found in back of textbook Pg 398 & 399

9. 3.5 Z-Score Tables

10. 3.5 Applying the Normal Distribution: Z-Scores
Example
The annual returns from a particular mutual fund are believed to be normally distributed. A sample of the last 15 years of historic returns are listed in the following table.
Determine the mean & standard deviation of the annual return.
What is the probability that an annual return will be:
At least 9%?
Negative?
What is the probability the investment will yield a return greater than 6%?

11. 3.5 Applying the Normal Distribution: Z-Scores
Year
Return (%) 1
7.2 6
19.3 11
6.4 2
12.3 7
12.2 12
27.0 3
17.1 8
-13.1 13
14.5 4
17.9 9
20.2 14
25.2 5
10.8 10
18.6 15
-0.5

12. 3.5 Applying the Normal Distribution:
Home Entertainment
Page 186 #1- 5, 7, 8, 10, 13, 17, 18