12.9 13.7 14.1 14.2 14.5 14.514.6 14.7 15.1 15.2 15.3 15.315.3 15.3 15.5 15.6 15.6 15.816.0 16.0 16.2 16.2 16.3 16.416.5 16.6 16.6 16.6 16.8 17.017.0 17.2 17.4 17.4 17.9 18.4

Do NOT glue (we’ll do that later)—simply type the data into List 1

[Before class begins—beginning of 2.2B]

2.2B INTRODUCTION TO NORMAL DISTRIBUTIONS

After this section, you should be able to…

PERFORM Normal distribution calculations using tables and/or technology

ASSESS Normality

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

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 , then the σstandardized variable

has the standard Normal distribution, N(0,1).

z x -

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.

How to Solve Problems Involving Normal Distributions

The heights of young American women are approx normally

distributed with mean 64.5 inches and stdev 2.5 inches.

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”?

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”?

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”?

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?

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.

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.

Check whether the data follow the 68-95-99.7 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.

Assessing Normality

12.9 13.7 14.1 14.2 14.5 14.514.6 14.7 15.1 15.2 15.3 15.315.3 15.3 15.5 15.6 15.6 15.816.0 16.0 16.2 16.2 16.3 16.416.5 16.6 16.6 16.6 16.8 17.017.0 17.2 17.4 17.4 17.9 18.4

 

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.

Use a Normal Probability Plot. • Sketch a Normal Probability Plot• Assess the NPP for approximate normality

Assessing Normality

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.

The heights of young American women are approx normally distributed with

mean 64.5 inches and stdev 2.5 inches.

1. % taller than 68”2. % shorter than 60”3. % between 60” and 68”

Let’s use the TI to calculate normal probabilities: