Download 2.2 More on Normal Distributions and Standard Normal Calculations

2.2 MORE ON NORMAL DISTRIBUTIONS AND STANDARD NORMAL CALCULATIONS
Standardizing
 We can compare ____________________________ that have different means and
standard deviations by standardizing.
z
x

The standardized value is called a ____________________.
x is the given value

This also allows you to find the __________________________ under a given part of
the curve.

Why standardizing is your friend

Standardizing ________________________________________________________
and your standard deviation to one and compare information in two normal distributions.
Who’s Taller? (relatively speaking)
 Verne is 67” tall. Assume the heights of women her age are normally distributed with a
mean μ = 64 inches and standard deviation σ = 2.5 inches.

Hank is 72” tall. Assume the heights of men his age are normally distributed with a
mean μ = 69.5 inches and standard deviation σ = 2.25 inches.
Since any normal curve can be standardized, we can find areas under the curve using one table,
Table A.
 It is very important to remember that Table A gives the __________________________
______________________________________________________________________!!

Also, standardized normal curves have a _____________________________________
______________________________________________________________________.

Use Table A to find the proportion of observations that have a z-score less than 1.4 (this
is 1.4 standard deviations from the mean).

Use Table A to find the proportion of observations greater than a z-score of -2.15.
Steps in Finding Normal Proportions
 Step 1: Make sure the variable of interest is from a _________________________!!!
Then draw a picture of the distribution and _________________________________

____________________________ with the values given (center and important points).
Step 2: Standardize x by using the formula.

Label your picture with the __________________________________________.

Step 3: Use Table A to _________________________________ under the curve.
 Step 4: State your conclusion in ____________________________________________
.
Now to the actual problems…
 A commonly used IQ “cut-off” score for AIG identification is 125. IQ scores on the
WISC-IV are normally distributed with a mean = 100 and a standard deviation = 15.
Find the proportion of people whose IQ score is at least 125.

IQs between 140 and 170 are commonly referred to as “moderately profoundly gifted.”
What proportion of the population have IQ scores between 140 and 170?

Scores on the SAT Verbal approximately follow the N(505,110) distribution. How high
must a student score to be in the top 10% of all students taking the SAT?
Caution about Test Items
 Many test items ask students to distinguish between types of density curves. Once the
hear the word, students have a tendency to call everything “normal.” Be careful!
What if they don’t tell me whether the data are from a normal population?
If you’re given the data, you have several ways to assess normality.
1. Start by looking __________________________________________________
_____________________________________________. Does the data appear
symmetrical, with most of the data being near the center?
2. And of course there is always the _____________________________…
3. Another method is to check the _______________________ rule. First, find the
mean and standard deviation. Then count what percent of the observations fall
within one standard deviation of the mean. Is it close to 68%? Repeat for 2 and 3
standard deviations away from the mean.
4. Normal Probability Plots!!!!!! (much easier)
i. method is to construct a ___________________________ using your
calculator.
ii. If the plot is __________________________________, it is safe to
assume the data are from a normal distribution.
Constructing Normal Prob. Plots
 Type your data in your calculator (it is probably already there, because I know you have
looked at your histogram or box-and-whisker plot!).

Go to StatPlot. Choose the last graph option. This represents Normal Probability.
Let’s look at page 108, problem 2.26.
For those of you who like your calculator…

To find the area under a normal curve…

2nd Vars, normal cdf

Normalcdf(min, max, μ, σ)

If you are looking at everything to the left of the max, then min = 1E-99.

If you are looking at everything to the right of the min, then max = 1E99.