Download 2.2 – Normal Distribution Notes

2.2 Normal Distributions & Standard Normal Distributions
Normal Distributions are density curves that are:
1.
2.
3.
The mean is labeled _____.
The standard deviation is labeled _____.
The 68-95-99.7 Rule:
TRY: The standard “IQ” scores of the 20 to 34 age group are normally distributed with
µ = 110 and σ = 25.
a. What percent of people in this age group has scores above 110?
b. What percent have scores above 160?
c. in what range do the middle 95% of all IQ scores lie?
A Standard Normal Distribution has mean = 0 and standard deviation 1. Every normal
distribution can be standardized by performing the following transformation:
If X is a score from N (µ ,σ ) distribution, then Z =
X −µ
is the standardized score for X.
σ
Once we standardize a score, we can easily analyze how that score relates to the mean
of the distribution. For example, a z-score of 1.2 means that score is 1.2 standard
deviations away from the mean. We can also use standard normal tables to answer
questions about the distribution.
We will use TABLE A (at the front of your textbook) to find the area under a standard
normal curve. The table entry for each value z is the area under the curve to the left of
z.
Example 1:
Suppose the distribution of SAT Math scores is N(870,35) .
a) What proportion of SAT Test scores is less than 850?
b) What proportion of SAT Test scores is between 800 and 900?
c) What proportion of SAT Test scores is greater than 890?
d) Find the score in the 60th percentile.
2.2 Assignment:
1. The scores of a reference population on the Wechsler Intelligence Scale for Children
(WISC) are normally distributed with µ = 100 and σ = 15.
a. What score would represent the 50th percentile? Explain.
b. Approximately what percent of the scores fall in the range from 70 to 130?
c. A score in what range would represent the top 16% of the scores?
2. Runner’s World reports that the times of the finishers in the New York City 10-km
run are normally distributed with a mean of 61 minutes and a standard deviation of 9
minutes.
a. Find the proportion of runners who take more than 70 minutes to finish. Draw a
sketch to show this proportion.
b. Find the proportion of runners who finish in less than 43 minutes. Draw a sketch
to show this proportion.
3. In a study of elite distance runners, the mean weight was reported to be 63.1
kilograms (kg), with a standard deviation of 4.8 kg.
a. Assuming that the distribution of weights is normal, sketch the density curve of the
weight distribution, with the horizontal axis marked in kilograms.
b. Jill scores 680 on the mathematics part of the SAT. The distribution of SAT scores in
a reference population is normally distributed with mean 500 and standard
deviation 100. Jack takes the ACT mathematics test and scores 27. ACT scores are
normally distributed with mean 18 and standard deviation 6. Find the standardized
scores for both runners.
c. Assuming that both tests measure the same kind of ability, who has the higher
score, and why?
4. Using Table A (table of standard normal probabilities) or your calculator, find the
proportion of observations from a standard normal distribution that satisfies each of the
following statements. In each case, sketch the normal curve and shade the area under
the curve that is the answer to the question.
a. Z < –1.5
b. 1.5 < Z < 0.8
5. The distribution of heights of adult American men is normal with mean 69 inches and
standard deviation 2.5 inches.
a. What percent of men are at least 6 feet (72in) tall?
b. What percent of men are between 5 feet (60in) and 6 feet tall?
c. How tall must a man be to be in the tallest 10% of all adult men?