1. Section 2.2 Standard Normal Calculations
AP Statistics
September 17th , 2014

2. AP Statistics, Section 2.1, Part 2
Comparing data sets
How do we compare results when they are measured on two completely different scales?
One solution might be to look at percentiles
What might you say about a woman that is in the 50th percentile and a man in the 15th percentile?
AP Statistics, Section 2.1, Part 2

3. Another way of comparing
Another way of comparing: Look at whether the data point is above or below the mean, and by how much.
Example: A man is 64 inches tall. The heights of men are normally distributed with a mean of 69 inches and standard deviation of 2.5 inches.
AP Statistics, Section 2.1, Part 2

4. Another way of comparing
Example: A man is 64 inches tall. The heights of men are normally distributed with a mean of 69 inches and standard deviation of 2.5 inches.
We can see that the man is below the mean, but by how much?
AP Statistics, Section 2.1, Part 2

5. Another way of comparing
Example: A man is 64 inches tall. The heights of men are normally distributed with a mean of 69 inches and standard deviation of 2.5 inches.
AP Statistics, Section 2.1, Part 2

6. AP Statistics, Section 2.1, Part 2
z-scores
The z-score is a way of looking at every data set, because each data set has a mean and standard deviation
We call the z-score the “standardized” score.
AP Statistics, Section 2.1, Part 2

7. AP Statistics, Section 2.1, Part 2
z-scores
Positive z-scores mean the data point is above the mean.
Negative z-scores mean the data point is below the mean.
The larger the absolute value of the z-score, the more unusual it is.
AP Statistics, Section 2.1, Part 2

8. AP Statistics, Section 2.1, Part 2
Using the z-table
We can use the z-table to find out the percentile of the observation.
A z-score of -2.0 is at the percentile.
AP Statistics, Section 2.1, Part 2

9. AP Statistics, Section 2.1, Part 2
Cautions
The z-table only gives the amount of data found below the z-score.
If you want to find the portion found above the z-score, subtract the probability found on the table from 1.
AP Statistics, Section 2.1, Part 2

10. Standardized Normal Distribution
We should only use the z-table when the distributions are normal, and data has been standardized
N(μ,σ) is a normal distribution
N(0,1) is the standard normal distribution
“Standardizing” is the process of doing a linear translation from N(μ,σ) into N(0,1)
AP Statistics, Section 2.1, Part 2

11. AP Statistics, Section 2.1, Part 2
Example
Men’s heights are N(69,2.5).
What percent of men are taller than 68 inches?
AP Statistics, Section 2.1, Part 2

12. AP Statistics, Section 2.1, Part 2
Step 1
State the problem in terms of x, an observed variable.
x is the height 68 inches.
AP Statistics, Section 2.1, Part 2

13. AP Statistics, Section 2.1, Part 2
Step 2
Standardize x to restate the problem in terms of a standard normal distribution.
AP Statistics, Section 2.1, Part 2

14. AP Statistics, Section 2.1, Part 2
Step 3
Find the required area under the standard normal curve. Remember, the total area is 1.
AP Statistics, Section 2.1, Part 2

15. AP Statistics, Section 2.1, Part 2
Step 4
Write your conclusion in the context of the problem.
About % of the population of men are taller than 68 inches.
AP Statistics, Section 2.1, Part 2

16. Working with intervals
What proportion of men are between 68 and 70 inches tall?
AP Statistics, Section 2.1, Part 2

17. Working with intervals
AP Statistics, Section 2.1, Part 2

18. Working with intervals
What proportion of men are between 68 and 70 inches tall?
AP Statistics, Section 2.1, Part 2

19. AP Statistics, Section 2.1, Part 2
Assignment
Exercises 2.19 – 2.25, The Practice of Statistics.
AP Statistics, Section 2.1, Part 2