1. Do NOT glue (we’ll do that later)—simply type the data into List 1
12.9
13.7
14.1
14.2
14.5
14.6
14.7
15.1
15.2
15.3
15.5
15.6
15.8
16.0
16.2
16.3
16.4
16.5
16.6
16.8
17.0
17.2
17.4
17.9
18.4
[Before class begins—beginning of 2.2B]

2. 2.2B Introduction to Normal Distributions

3. After this section, you should be able to…
PERFORM Normal distribution calculations using tables and/or technology
ASSESS Normality

4. The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.
approx N(64.5, 2.5)inches
Remind them that mean of all the z’s = mean of the observations (64.5) minus what you subtracted from all the observations (64.5)…and that’s how you get mean(z’s) = 0. Likewise, stdv of all the z’s = stdv of all the observations (2.5) divided by (2.5)…and that’s how you get stdv(z’s) = 1

5. The Standard Normal Distribution
All Normal distributions can be transformed into one, STANDARD Normal distribution by measuring in units of size σ from the mean µ as center.

6. has the standard Normal distribution, N(0,1).
Definition:
The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable
has the standard Normal distribution, N(0,1).

7. Express the problem in terms of the observed variable x.
How to Solve Problems Involving Normal Distributions
Express the problem in terms of the observed variable x.
Draw a picture of the distribution and shade the area of interest under the curve.
Perform calculations.
Standardize x in terms of z.
Use Table A to find the required area under the standard Normal curve.
Write your conclusion in context.

8. The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.

9. The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.
1. What % are taller than 68”?

10. The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.
2. What % are shorter than 60”?

11. The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.
3. What % are between 60”and 68”?

12. The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.
4. How tall would a person in the top ten percent have to be?

13. Assessing Normality
(I am going to say something like this, but you don’t need to copy this down )
The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures are based on the assumption that the population is approximately Normally distributed. Consequently, we need a strategy for assessing Normality.

14. Assessing Normality Plot the data! Plot the data! PLOT THE DATA!
Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell- shaped. (This is step 1.)
Check whether the data follow the rule.
Count how many observations fall within one, two, and three standard deviations of the mean and check to see if the percents are close to the 68%, 95%, and 99.7% targets for a Normal distribution.

15. 12.9
13.7
14.1
14.2
14.5
14.6
14.7
15.1
15.2
15.3
15.5
15.6
15.8
16.0
16.2
16.3
16.4
16.5
16.6
16.8
17.0
17.2
17.4
17.9
18.4
See if the data fits “into” the rule.
This data is in list 1….

16. Assessing Normality  Plot the data. (Do I need to say it thrice?)
Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell- shaped. (Step 1….remember?)
Use a Normal Probability Plot.
Sketch a Normal Probability Plot
Assess the NPP for approximate normality

17. The Normal Probability Plot
Plots each observation against its z-score
If the points on a NPP lie close to a straight line, the plot indicates that the data are approx Normal.
Systematic deviations from a straight line indicate a non-Normal distribution.
Outliers appear as points that are far away from the overall pattern of the plot.

18. Example of a histogram and what it would look like as an NPP.

22. Let’s use the TI to calculate normal probabilities:
The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.
% taller than 68”
% shorter than 60”
% between 60” and 68”

24. cdf cumulative density function

25. (lower, upper, mean, stdv)

26. So, using normalpdf(lower, upper, µ,σ)
The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.
% taller than 68”
% shorter than 60”
% between 60” and 68”