1. I can find any probability…from a normal distribution
EQ: How can any normal distribution be standardized?

2. Standardizing
Any normal distribution can be converted to the standard normal using the following:
The value is referred to as a “z-score” and represents how many standard deviations the observation is from the mean.

3. Former NBA star Michael Jordan is 78 in
Former NBA star Michael Jordan is 78 in. tall, while WNBA player Rebecca Lobo is 76 in. tall. Men’s heights have a mean of 69 in. and a standard deviation of 2.8 in. Women’s heights have a mean of 63.6 in. and a standard deviation of 2.5 in. Which player is relatively taller?
66.2
71.8
74.6
77.4 69
63.4
60.6
Mens Heights
61.1
66.1
68.6
71.1
63.6
58.6
56.1
Womens Heights

4. Locate each player MJ Height = 78 inches Mens heights: N(69,2.8) 65.2
71.8
74.6
77.4 69
62.4
59.6
Lobo Height = 76 inches
Womens heights: N(63.6,2.5)
61.1
66.1
68.6
71.1
63.6
58.6
56.1

5. Locate them on the standard normal-1 1 2 3 -2 -3
3.2
4.96

6. Conclusion
Rebecca Lobo is relatively taller than Michael Jordan. Rebecca is 4.96 standard deviations from the mean of female players where as Michael is 3.2 standard deviations from the mean men’s height.

7. Types of questions A. Find the probability of an event:
Write a probability statement in terms of x
Draw a diagram and shade
Find the z-score and probability.
Write an answer in context.

8. Example
Mugsy Bogues is 63 in. tall. What percent of people are taller than him?
Recall men’s heights: N(69,2.8)
P(x>63)
2. Draw a picture
3. Find the z-score and
probability
P( z > )=.9839
% of men are taller than Mugsy Bogues.

9. Types of questions B Find a cut off point Draw a diagram
Find the corresponding z-score
Write an equation for the z-score
Solve for the missing piece

10. Example What is the height for a man at the 70th percentile ?
70th percentile means 70% of men are shorter.
1. Draw a picture
2. Find the z-score
Inverse Norm of .7
InvNorm(.7) = = z score
3. Write an equation and solve

11. Notes about InvNorm Only calculates to the left of a given z-score
Do 1- area to get a right shaded z-score

12. Example What are the heights of the tallest 60% of men ?
Same as Z-score as shortest 40%
Tallest 60%
Use invNorm of .4 instead of .6

13. What are the heights of the tallest 60% of NBA players?
2. Find the z-score
InvNorm (.4)
=-.2533
3. Write an equation
60% of men are taller than inches

14. Example
Consider babies born in the “normal” range of 37 to 43 weeks of gestation. Extensive data supports the assumption that for such babies born in the U.S., birth weight is normally distributed with a mean of 3432 g, and a standard deviation of 482 g.

15. Baby Weights ~ N(3432, 482)
1. What is the probability that the birth weight of a randomly chosen baby of this type is less than 4000 g? 2. What is the probability that the birth weight of a randomly chosen baby of this type is more than 3000 g? 3. What is the probability that the birth weight will be between 3000 and 4000 g?

16. Baby Weights ~ N(3432, 482)
4. What weight would a baby need to weigh in order to be in the 25th percentile? 5. A mother had a baby that weighed 4900 grams. She is embarrassed and does not want to report that she has an unusually large baby. If the mean remains 3432 grams, what would the standard deviation need to be so that her baby was a “usual” weight. Let’s consider all usual weight babies to fall between the 25th and 75th percentiles. What would the standard deviation need to be so that her baby is at the upper bound? 6. Repeat question 6 but allow the mean to change and the standard deviation to remain fixed at 482 grams.