Chapter 6: Continuous Probability
Distributions
Slide set to accompany "Statistics Using Technology" by Kathryn Kozak (Slides by David H Straayer) is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at
http://www.tacomacc.edu/home/dstraayer/published/Statistics/Book/StatisticsUsingTechnology112314b.pdf.
6.1 Uniform Distribution
If you have a situation where the probability is
always the same, then this is known as a uniform
distribution. An example would be waiting for a
commuter train. The commuter trains on the Blue
and Green Lines for the Regional Transit Authority
(RTA) in Cleveland, OH, have a waiting time during
peak hours of ten minutes ("2012 annual report,"
2012). If you are waiting for a train, you have
anywhere from zero minutes to ten minutes to
wait. Your probability of having to wait any number
of minutes in that interval is the same.
Uniform Distribution Graph
Probability of an interval
Questions about a uniform
distribution: our RTA wait time
a) State the random variable.
b) Find the probability that you have to wait
between four and six minutes for a train.
c) Find the probability that you have to wait
between three and seven minutes for a train.
d) Find the probability that you have to wait
between zero and ten minutes for a train.
e) Find the probability of waiting exactly five
minutes.
Why study uniform distributions?
• Computer random number generators are used
for a lot of things. They usually generate a
random number in the interval [0.0,1.0)
• They introduce the idea of probability as an area
under a (straight or curved) line.
• They can be used to construct “toy” problems for
a statistics class.
• For example, the sum of two uniform random
numbers makes a slightly more interesting (but
still calculable) geometry for areas.
6.2: Graphs of the Normal Distribution
“Normal” has a special meaning here.
Many real life problems produce a histogram
that is a symmetric, unimodal, and bell-shaped
continuous probability distribution. For
example: height, blood pressure, and cholesterol
level. However, not every bell shaped curve is a
normal curve. In a normal curve, there is a
specific relationship between its “height” and its
“width.”
Typical Graph of a Normal Curve
Characteristics of a normal curve
• The center, or the highest point, is at the
population mean, .
• The transition points (inflection points) are the
places where the curve changes from a “hill” to a
“valley”. The distance from the mean to the
transition point is one standard deviation, .
• The area under the whole curve is exactly 1.
Therefore, the area under the half below or
above the mean is 0.5.
“” and “<“, what’s the big deal?
• In algebra, we make a big deal about of the
difference between a  x  b and a < x < b.
• In terms of probabilities, there is no
difference.
• For example, the probability of being taller
than 6 feet tall, and the probability of being 6
feet tall or taller, is exactly the same.
• Because no one is exactly 6 feet tall!
Probability of an event as an area
Empirical Rule
The Empirical Rule for any normal distribution:
• Approximately 68% of the data is within one
standard deviation of the mean.
• Approximately 95% of the data is within two
standard deviations of the mean.
• Approximately 99.7% of the data is within
three standard deviations of the mean.
(This like Chebyshev’s rule, but more specific)
Empirical Rule
z-score
• The z-score is normally distributed, with
a mean of 0 and a standard deviation of
1. It is known as the standard normal
curve.
• Before modern technology, we had to
calculate a z-score to look up in a table.
Standard Normal Curve
Z
TI Technology
• The command on the TI-83/84 is in the DISTR
menu and is normalcdf(. You then type in the
lower limit, upper limit, mean, standard
deviation in that order and including the
commas.
• Having to use both lower limit and upper limit
is sometimes a convenience, and often a
nuisance. Especially when there is no lower
limit or no upper limit.
The “Bell” program
• I’ll help transfer the “Bell” program into your
calculator.
• It leads you step-by-step through normal (and
T, but we’ll get to that later) probability
calculations.
This program is merely a “wrapper”
around the TI’s built-in statistical
functions. It doesn’t calculate
anything that the TI can’t do already,
but it provides a user interface that
allows you to focus on the statistical
implications of what they are
calculating. It helps avoid some of
the eccentricities of the user
interface of the built-in functions. It
also provides the functionality of the
Inverse T (InvT) function missing
from the TI-83. It runs on all versions
of the TI-83 and TI-84, as well as the
TI-Nspire.
Running Bell
Let’s assume you are
just learning about the
Normal (Z) curve, and
want to find an area
under a region of the
curve. For example, a
problem may ask you to
find P(Z>2.0), the
probability that Z is
greater than 2.0.
• Stick with NORMAL
for now.
• Sometime we work
with “Z” for the
standard normal
curve, sometimes we
use “X” coordinates
when the mean  0
or s.d.  1.0
• If there is one boundary, you may
be interested in the area to the
left, or the area to the right.
• If you have two boundaries, you
are usually interested in the area
between (inside), but sometimes
you want the area outside.
• After telling BELL which area you
want, give the boundary or
boundaries.
• The result is stored in a memory
location named “P”, which you
can use in later calculations.
Using X instead of Z
In this example, we’ll explore
“real-world” numbers. For
example, college males have
heights normally distributed
with a mean of about 70
inches and a standard
deviation of 2.8 inches. What
is the probability that a
randomly chosen blind date is
over 6 feet (72 inches) tall?
There is about a
23.7% chance
that a randomlychosen male
college student is
over six feet tall.
Once again, the
result is saved in
a named memory
“P”.
“Going the other way”
Sometimes we know an area/probability, and
we want to know what X or Z value traps that
area to the left or right. Sometimes we even
have an “inside” area, centered around the
mean, and we want to know what values trap
that area. If these are Z values, we sometimes
refer to these as Z* or Zc or Z/2. We often refer
to numbers like this as critical values.
Confirm Empirical rule 95% is 2 
Do 2 ’s trap 95%?
• Pretty close.
• Actually, it is
more like 1.96
standard
deviations.
• “Close Enough!”
The length of a human pregnancy is normally
distributed with a mean of 272 days with a standard
deviation of 9 days
a) State the random variable
b) Find the probability of a pregnancy lasting more than 280
days.
c) Find the probability of a pregnancy lasting less than 250
days.
d) Find the probability that a pregnancy lasts between 265
and 280 days.
e) Find the length of pregnancy that 10% of all pregnancies
last less than.
f) Suppose you meet a woman who says that she was
pregnant for less than 250 days. Would this be unusual
and what might you think?
SAT: =514, =117, normally distributed
a) State the random variable.
b) Find the probability that a person has a
mathematics SAT score over 700.
c) Find the probability that a person has a
mathematics SAT score of less than 400.
d) Find the probability that a person has a
mathematics SAT score between a 500 and a
650.
e) Find the mathematics SAT score that represents
the top 1% of all scores.
Section 6.4: Assessing Normality
1. What is the source of variation in the
population? If it is lots of different factors, each
adding or subtracting, then the distribution is
probably normal.
2. Histogram: should roughly be bell-shaped
3. Outliers: should not be more than 1. Use the
1.5*IQR rule.
4. Normal Probability Plot: if the points lie close to
a straight line, data is approximately normal.
Make Histogram
• Enter Data
• Clear out old plots! Both STAT plots and Y=
plots.
• Set up a plot, choose histogram, identify the
data.
• Zoom 9
Use Boxplot to identify outliers
• Don’t forget to clear out old plots!
• Create a Boxplot
• Chose the fourth type, not the fifth type
suggested by the book. It uses the 1.5*IQR
rule to show outliers.
• Zoom 9
Normal Probability Plot
• This is the sixth plot in the menu of STAT PLOT types.
• You need to set up the correct window on which to graph.
Click on WINDOW. You need to set up the settings for the x
variable. Xmin should be . Xmax should be 4. Xscl should
be 1. Ymin and Ymax are based on your data, the Ymin
should be below your lowest data value and Ymax should
be above your highest data value. Yscl is just how often you
would like to see a tick mark on the y-axis.
• Press GRAPH
• If close to a straight line, this suggests normality.
Kiama Blowhole
Skewed right, not normal
Outliers: two, suspicious.
Conclusion: not normal
Curved, not straight line
IQ scores
Look bell shaped
No outliers
Conclusion, probably normal
looks linear
Section 6.5: Sampling Distribution
and the Central Limit Theorem
• Statistical Inference: to make accurate
decisions about parameters from statistics
• Sampling Distribution: how a sample statistic
is distributed when repeated trials of size n
are taken.
Sampling Distribution
• Suppose you have a random variable that has
a population mean, , and a population
standard deviation, . If a sample of size n is
taken, then the sample mean 𝑥, has a mean
𝜎
𝜇𝑥 = 𝜇 and standard deviation of 𝜎𝑥 = .
𝑛
Central Limit Theorem
• If the random variable has a normal
distribution, 𝑥 will be normally distributed.
• If the random variable has any distribution,
the distribution of 𝑥 will become normally
distributed as n increases.
• How big does n have to be? It depends on the
shape of the original distribution, but around
30 the sampling distribution usually gets
pretty close to normal.
The age that American females first have intercourse is on
average 17.4 years, with a standard deviation of
approximately 2 years. This random variable is not normally
distributed, though it is somewhat mound shaped.
a) State the random variable.
b) Suppose a sample of 35 American females is taken.
Find the probability that the mean age that these 35
females first had intercourse is more than 21 years.
Could the population mean have increased from the 17.4
years that was stated in the article? Could the sample not
have been random, and instead have been a group of
women who had similar beliefs about intercourse? These
questions, and more, are ones that you would want to
ask as a researcher