1. Other Normal Distributions
Lecture 21
Section 6.3.1
Fri, Oct 15, 2004

2. Other Normal Curves
The standard normal table and the TI-83 function normalcdf are for the standard normal distribution only.
If we are working with a different normal distribution, say N(30, 5), then how can we find areas under the curve?

3. Other Normal Curves
For example, if X is N(30, 5), what is the area to the left of 35? 15 20 25 30 35 40 45

4. Other Normal Curves
For example, if X is N(30, 5), what is the area to the left of 35? 15 20 25 30 35 40 45

5. Other Normal Curves
For example, if X is N(30, 5), what is the area to the left of 35? ? 15 20 25 30 35 40 45

6. Other Normal Curves
To determine the area, we need to find out how many standard deviations 35 is above average.
Since  = 30 and  = 5, we find that 35 is 1 standard deviation above average.
Thus, we may look up 1.00 in the standard normal table and get the correct area.

7. Other Normal Curves For example, what is the area to the left of 35?
0.8413 X 15 20 25 30 35 40 45 Z -3 -2 -1 1 2 3

8. Z-Scores
Z-score, or standard score, of an observation – The number of standard deviations from the mean to the observed value.
Compute the z-score of x as or
Therefore

9. Areas Under Other Normal Curves
If a population has a normal distribution, then what is the distribution of the z-scores of its members?
Let X be N(, ).
Then (X – )/ is N(0, 1).
That is,

10. Example
Let X be N(30, 5).
What proportion of values of X are below 38?
Compute z = (38 – 30)/5 = 8/5 = 1.6.
Find the area to the left of 1.6 under the standard normal curve.
Answer:
Therefore, 94.52% of the values of X are below 38.

11. Let’s Do It!
Let’s Do It! 6.2, p. 333 – IQ Scores.

12. Standard Normal Percentiles
Given a value of Z, we know how to find the area to the left of that value of Z.
Z  Area to the left
The problem of finding a percentile is exactly the reverse: Area to the left  Z
Given the area to the left of a value of Z, find that value of Z?
That is, given the percentage, find the percentile.

13. Standard Normal Percentiles
What is the 90th percentile of Z?
That is, find the value of Z such that the area to the left is
Look up as an entry in the standard normal table.
Read the corresponding value of Z.
Z = 1.28.

14. Practice Find the 99th percentile of Z. Find the 1st percentile of Z.
Find Q1 and Q3 of Z.
What value of Z cuts off the top 20%?
What values of Z determine the middle 30%?

15. TI-83 – Standard Normal Percentiles
To find a standard normal percentile on the TI-83,
Press 2nd DISTR.
Select invNorm.
Enter the percentile as a decimal (area).
Press ENTER.

16. TI-83 – Standard Normal Percentiles
invNorm(0.99) =
invNorm(0.01) =
invNorm(0.50) = 0.
Q1 = invNorm(0.25) =
Q3 = invNorm(0.75) =
invNorm(0.80) =
invNorm(0.35) =
invNorm(0.65) =

17. Normal Percentiles
To find a percentile of a variable X that is N(, ),
Find the percentile for Z.
Use the equation X =  + Z to find X.

18. Example Let X be N(30, 5). Find the 99th percentile of X.
The 99th percentile of Z is
Therefore, X = 30 + (2.236)(5) =
99% of the values of X are below

19. TI-83 – Normal Percentiles
Use the TI-83 to find the standard normal percentile and then use the equation
X =  + Z.
Or, use invNorm and specify  and .
invNorm(0.95, 30, 5) = 38.2.