George Mason University Probability and Statistics Worksheet
please do the following
#11.15 – 11.24, 11.28, 11.30, 11.31, 11.35
The Normal Distribution
Chapter 11
Normal Curves
• Normal curves are symmetric, single-peaked, and bell-shaped distributions.
• The exact density curve for a particular Normal distribution is described by giving its
mean, 𝜇, and its standard deviation, 𝜎.
• The mean of a Normal distribution is at the center, and the standard deviation is the
distance from the center to the change-of-curvature points on either side.
Not all data are
Normally distributed.
𝑓 𝑥 =
1
2𝜋
1 𝑥−𝜇 2
−2 𝜎
𝑒
Here means are the same (µ = 15)
whereas standard deviations are
different (σ = 2, 4, and 6).
Here means are different
(µ = 10, 15, and 20) whereas standard
deviations are the same (σ = 3).
Human heights, by gender, can be modeled
quite accurately by a Normal distribution.
Guinea pigs survival times after inoculation of a
pathogen are clearly not a good candidate for a
Normal model!
The 68-95-99.7 Rule (Empirical Rule)
All normal curves, 𝑁(µ, 𝜎), share the same properties:
•
•
•
•
About 68% of all observations are within 1 standard deviation (σ) of the mean (µ).
About 95% of all observations are within 2 σ of the mean µ.
Almost all (99.7%) observations are within 3 σ of the mean.
To obtain any other area under a Normal curve, use either technology or Table B.
Example
The distribution of heights of young women age 18 to 24 is approximately normal
with 𝜇 = 64.5inches and standard deviation of 𝜎 = 2.5inches. According to the
Empirical Rule, the heights of the middle 95% of young women are between what
two values?
𝜇 − 2 𝜎 = 64.5 − 2 2.5 = 64.5 − 5 = 59.5 𝑖𝑛𝑐ℎ𝑒𝑠
𝜇 + 2 𝜎 = 64.5 + 2 2.5 = 64.5 + 5 = 69.5 𝑖𝑛𝑐ℎ𝑒𝑠
The middle 95% of women’s heights is between 59.5 inches and 69.5 inches.
Example
A height of 62 inches is one standard deviation below the mean. What is the
probability that a woman is taller than 62 inches?
The probability that a randomly chosen female between 18 to 24 years
old is taller than 62 inches is approximately 84%.
Z-Score
If 𝑥 is an observation from a distribution that has mean 𝜇 and standard deviation,
𝜎, the standardized value of 𝑥 is:
𝑥−𝜇
𝑧=
𝜎
A standardized value is often called a z-score.
A z-score tells us how
many standard deviations
the original observation
falls away from the mean,
and in which direction.
Example
The heights of young women aged 18 to 24 are approximately Normal, with 𝜇 =
64.5 inches and standard deviation of 𝜎 = 2.5 inches. Determine a woman’s
standardized height if she is 70 inches tall, similarly for a woman who is 60 inches
tall (5ft).
70 − 64.5
𝑧=
= 2.2
2.5
60 − 64.5
𝑧=
= −1.8
2.5
Standard Normal Distribution
The standard Normal distribution is the Normal distribution 𝑁(0,1) 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
𝑥−𝜇
𝑧=
𝜎
has the standard Normal distribution.
Areas under a Normal curve represent proportions (frequencies) of observations
from that Normal distribution.
The cumulative probability for a value 𝑥 in a distribution is the proportion of
observations in the distribution that lie at or below 𝑥.
Example
Osteoporosis is a condition in which the bones become brittle to the breaking
point due to loss of minerals. To diagnose osteoporosis, bone mineral density
(BMD) is measured and typically reported as a unitless score, which is designed so
that the mean BMD score of healthy young adults of the same sex is zero. The
WHO defining criterion for osteoporosis is a BMD score below -2.5.
Women in their 70s tend to have much lower BMD. In this population, BMD scores
are approximately Normally distributed with a mean of -2 and a standard deviation
of 1. Find the probability that a randomly selected woman in her 70s would have
osteoporosis.
Example
Osteopenia is a condition defined by less than ideal bone density, corresponding to
BMD scores between -2.5 and -1. We want to find what percent of women in their
70s have osteopenia.
53.28% of women in their 70s have
osteopenia.
Percentile
The value such that some percent 𝑝 of the observations in a distribution lie
below it is called the 𝒑th percentile.
Example:
The hatching weights of commercial chickens can be modeled accurately using a
Normal distribution with mean 𝜇 = 45 grams and standard deviation 𝜎 = 4 grams.
What is the third quartile of the distribution of hatching weights?
Example
The lengths of human pregnancies from conception to birth (measured in days)
follow approximately the 𝑁(266,16) distribution. How long are the longest 10% of
pregnancies?
Pregnancies longer than
286 days are in the top
10% of pregnancy
lengths.
Example
The blood cholesterol levels of men aged 55 to 64 are approximately normal
with mean 222 mg/dL and standard deviation 37 mg/dL.
What range of values corresponds to the 10% highest cholesterol levels?
A) > 175 B) > 247
C) > 269
D) > 288
Tricks of Symmetry
Because of the curve’s symmetry, there are two ways of finding the area under
𝑁(0,1) curve to the right of a z-value.
Normal Quintile Plots
• One way to assess if a data set has an approximately Normal distribution is to
plot the data on a Normal quantile plot.
– The data points are ranked and the percentile ranks are converted to zscores. The z-scores are then used for the horizontal axis and the actual
data values are used for the vertical axis.
• If the data have approximately a Normal distribution, the Normal quantile
plot will have roughly a straight-line pattern.
Normal Quintile Plot Examples
Roughly normal
(~ straight-line pattern)
Right skewed
(most of the data points are short
survival times, while a few are longer
survival times)
