1. 2.2 Standard Normal Calculations
AP Statistics
2.2 Standard Normal Calculations

2. Learning Objectives: Calculate and interpret a z-score
Standardize data to a standard normal distribution
Calculate the proportion of observations under an interval on a normal curve

3. All normal distributions are the same if we measure in units of size σ about the mean μ as the center. Changing to these units is called:
standardizing

4. Standardizing and z-scores
If x is an observation from a distribution that has a mean μ and standard deviation σ, the standardized value of x is:
***A standardized value is called a z-score***

5. Ex: The heights of young women are approximately normal with μ=64
Ex: The heights of young women are approximately normal with μ=64.5 inches and σ=2.5 inches.
The standardized height is:
A women’s standardized height is the
The # of standard deviation by which the height differs from the mean height of all young women.

6. What is the standardized height of a woman who is 68 inches tall?
What does this mean?
1.4 standard deviation above the mean height

7. What about a woman who is 5 feet tall?
What does this mean?
1.8 standard deviation below the mean height

8. What does z-scores do?
Make all normal distributions into a single distribution. (with mean =0 and a standard deviation of 1)

9. Standard Normal Distribution
The standard normal distribution is the normal distribution N(0,1) with μ= 0 and σ= 1.

10. Ex: SAT versus ACT- Who did better?
Compare: Amy scored 680 on the math section of the SAT. The mean score is 500 with a standard deviation of Tom scores 27 on the ACT. The test is normally distributed with a mean of 18 and a standard deviation of 6.
Find the standardized scores for both students.
Amy: Tom:

11. Show both scores on a standard normal curve and determine who did better!

12. To get on calc: 2nd/VARS/2/Normalcdf Normalcdf(1.6,1000,0,1)
Ex: Find the proportion of observations from a standard normal distribution that are greater than 1.6?
Solution:
P(z>1.6)=
To get on calc: 2nd/VARS/2/Normalcdf
Normalcdf(1.6,1000,0,1)

13. Ex: Find the proportion of observation from a standard normal distribution that are less than -2.35?
Solution:
P(z<-2.35)=
To get on calc: 2nd/VARS/2/Normalcdf
Normalcdf(-1000,-2.35,0,1)

14. Step to Finding Normal Proportions
Step 1: Draw a curve and shade your area of
interest
Step 2: Write in terms of x, then convert to a
z-score.
P(x>___ ) = P(z>___ )
Step 3: Find the probability
Step 4: Write a conclusion in CONTEXT OF THE PROBLEM!!!!!

15. About 1% of 14 yr. old boys have cholesterol that is above 240 mg/dl.
Ex: Blood cholesterol of 14 year old boys have a mean of 170 mg/dl with a standard deviation of 30 mg/dl. Any levels above 240 mg/dl require medical attention. What percent of 14 year old boys require medical attention?
P(x>240)=P(z>2.3333)=0.0098
About 1% of 14 yr. old boys have cholesterol that is above 240 mg/dl.

16. Assessing Normality
Method 1: Given data, make a graph (histogram ,stem plot, etc….)
Method 2: Given statistics (no actual data), assume normality.