Transcript Lecture Notes_Cont Prob Model - Department of Statistics and

CHAPTER 8_B
• PROBABILITIES FOR
CONTINUOUS RANDOM
VARIABLES
• THE NORMAL
DISTRIBUTION
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REMINDER: Continuous Random Variable
 A continuous random variable has an infinite
continuum of possible values in an interval.
 Examples are: time, age, and size measures such as
height and weight.
 Continuous variables are usually measured in a
discrete manner because of rounding.
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Probability Distribution of a Continuous
Random Variable
A continuous random variable has possible values that
form an interval. Its probability distribution is specified
by a curve.
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
Each interval has probability between 0 and 1.

The interval containing all possible values has
probability equal to 1.
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Probability Distribution of a Continuous
Random Variable
Smooth curve approximation
Figure 8.2 Probability Distribution of Commuting Time. The area under the curve for
values higher than 45 is 0.15. Question: Identify the area under the curve represented by
the probability that commuting time is less than 15 minutes, which equals 0.29.
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Useful Probability Relationships
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Useful Probability Relationships
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Useful Probability Relationships
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Useful Probability Relationships
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Bell-Shaped Distributions
Probabilities for Bell – Shaped
Distributions or Continuous
Random Variables
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Normal Distribution
The normal distribution is symmetric, bell-shaped and
characterized by its mean  and standard deviation  .
 The normal distribution is the most important
distribution in statistics.
 Many distributions have an approximately normal
distribution.
 The normal distribution also can approximate
many discrete distributions well when there are a
large number of possible outcomes.
 Many statistical methods use it even when the
data are not bell shaped.
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Normal Distribution
Normal distributions are
 Bell shaped
 Symmetric around the mean
The mean ( ) and the standard deviation ( ) completely
describe the density curve.
 Increasing/decreasing  moves the curve along the
horizontal axis.
 Increasing/decreasing  controls the spread of the
curve.
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Normal Distribution
Within what interval do almost all of the men’s heights
fall? Women’s height?
Figure 8.4 Normal Distributions for Women’s Height and Men’s Height. For each different
combination of  and  values, there is a normal distribution with mean  and standard
deviation  . Question: Given that  = 70 and  = 4, within what interval do almost all of the
men’s heights fall?
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Empirical Rule or 68-95-99.7 Rule for
Any Normal Curve
≈ 68% of the observations fall within one standard deviation of the mean.
≈ 95% of the observations fall within two standard deviations of the mean.
≈ 99.7% of the observations fall within three standard deviations of the mean.
Figure 8.5 The Normal Distribution. The probability equals approximately 0.68 within
1 standard deviation of the mean, approximately 0.95 within 2 standard deviations,
and approximately 0.997 within 3 standard deviations. Question: How do these
probabilities relate to the empirical rule?
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Empirical Rule or 68 – 95 – 99.7% Rule
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Example : 68-95-99.7% Rule
Heights of adult women can be approximated by a normal
distribution,   65 inches;   3.5 inches
68-95-99.7 Rule for women’s heights:
68% are between 61.5 and 68.5 inches
 [     65  3.5]
95% are between 58 and 72 inches
 [   2  65  2(3.5)  65  7]
99.7% are between 54.5 and 75.5 inches
 [   3  65  3(3.5)  65  10.5]
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Z-Scores and the Standard Normal
Distribution
The z-score for a value x of a random variable is the
number of standard deviations that x falls from the mean.
z
x 

A negative (positive) z-score indicates that the value is
below (above) the mean.
Z-scores
can be used to calculate the probabilities of a

normal random variable using the normal tables in the
back of the book.
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Standard Score (z – score)
The formula for converting any value x to a z-score is
.
Value  Mean
x
z

Standard deviation

A z-score measures the number of standard
deviations that a value falls from the mean
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Shifting and Rescaling Data
Shifting data:
 Adding (or subtracting) a constant to every data
value adds (or subtracts) the same constant to
measures of position.
 Adding (or subtracting) a constant to each value will
increase (or decrease) measures of position:
center, percentiles, max or min by the same
constant.
 Its shape and spread - range, IQR, standard
deviation - remain unchanged.
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Shifting and Rescaling Data
The following histograms show a shift from men’s
actual weights to kilograms above recommended
weight:
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Shifting and Rescaling Data
Rescaling data:
 When we multiply (or divide) all the data values
by any constant, all measures of position (such
as the mean, median, and percentiles) and
measures of spread (such as the range, the IQR,
and the standard deviation) are multiplied (or
divided) by that same constant.
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Shifting and Rescaling Data
The men’s weight data set measured weights in
kilograms. If we want to think about these weights in
pounds, we would rescale the data:
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Examples: Midterm Exam II Practice Sheet
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Z-Scores and the Standard Normal
Distribution
A standard normal distribution has mean   0 and
standard deviation   1 .
When a random variable has a normal distribution
and its values are converted to z-scores by
subtracting the mean and dividing by the standard
deviation, the z-scores follow the standard normal
distribution.
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Table Z: Standard Normal Probabilities
Table Z enables us to find normal probabilities.
 It tabulates the normal cumulative probabilities
falling below the point   z .
To use the table:
 Find the corresponding z-score.
 Look up the closest standardized score (z) in the
table.
 First column gives z to the first decimal place.
 First row gives the second decimal place of z.
 The corresponding probability found in the body of
the table gives the probability of falling below the
z-score.
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Finding Probabilities Using The Standard
Normal Table (Table Z)
The figure shows us how to find the area to the left when
we have a z-score of 1.80:
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Example: Using Table Z
Find the probability that a normal random variable takes
a value less than 1.43 standard deviations above  ;
P( z  1.43)  0.9236
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Example: Using Table Z
Find the probability that a normal random variable
assumes a value within 1.43 standard deviations of  .
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
Probability below 1.43  0.9236

Probability below 1.43  0.0764

P(1.43  z  1.43)  0.9236  0.0764  0.8472
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Example: Using Table Z
Figure 8.7 The Normal Cumulative Probability, Less than z Standard Deviations
above the Mean. Table Z lists a cumulative probability of 0.9236 for z  1.43, so
0.9236 is the probability less than 1.43 standard deviations above the mean of any
normal distribution (that is, below   1.43 ). The complement probability of 0.0764 is
the probability above   1.43 in the right tail.
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From Percentiles to z - Scores
Sometimes we start with areas and need to find the
corresponding z-score or even the original data value.
Example: What z-score represents the first quartile in
a Normal model?
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From Percentiles to z - Scores
Look in Table Z for an area of 0.2500.
The exact area is not there, but 0.2514 is pretty close.
This figure is associated with z = -0.67, so the first
quartile is 0.67 standard deviations below the mean.
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How Can We Find the Value of z for a
Certain Cumulative Probability?
To solve some of our problems, we will need to find
the value of z that corresponds to a certain normal
cumulative probability.
To do so, we use Table A in reverse.
 Rather than finding z using the first column
(value of z up to one decimal) and the first row
(second decimal of z).
 Find the probability in the body of the table.
 The z-score is given by the corresponding
values in the first column and row.
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How Can We Find the Value of z for a
Certain Cumulative Probability?
Example: Find the value of z for a cumulative probability of 0.025.
Look up the cumulative probability of 0.025 in the body of Table A.
A cumulative probability of 0.025
corresponds to z  1.96 .
Thus, the probability that a normal
random variable falls at least
1.96 standard deviations
below the mean is 0.025.
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Are You Normal? Normal Probability Plots
When you actually have your own data, you must
check to see whether a Normal model is reasonable.
Looking at a histogram of the data is a good way to
check that the underlying distribution is roughly
unimodal and symmetric.
A more specialized graphical display that can help you
decide whether a Normal model is appropriate is the
Normal probability plot.
If the distribution of the data is roughly Normal, the
Normal probability plot approximates a diagonal
straight line. Deviations from a straight line indicate
that the distribution is not Normal.
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Are You Normal? Normal Probability Plots
Nearly Normal data have a histogram and a Normal
probability plot that look somewhat like this example:
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Are You Normal? Normal Probability Plots
A skewed distribution might have a histogram and
Normal probability plot like below. In such cases it is
unwise to use the Normal Model.
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Example: Comparing Test Scores That
Use Different Scales
Z-scores can be used to compare observations from
different normal distributions.
Picture the Scenario:
There are two primary standardized tests used by
college admissions, the SAT and the ACT.
You score 650 on the SAT which has  500 and  100
and 30 on the ACT which has   21.0 and   4.7 .
How can we compare these scores to tell which score is
relatively higher?
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Using Z-scores to Compare Distributions

Compare z-scores:
SAT: z 
650  500
1.5
100
30  21
 1.91
ACT: z 

4.7
Since your z-score is greater for the ACT, you

performed relatively better on this exam.
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SUMMARY: Using Z-Scores to Find Normal
Probabilities or Random Variable x Values
 If we’re given a value x and need to find a probability,
convert x to a z-score using z  ( x   ) /  , use a table of
normal probabilities (or software, or a calculator) to get a
cumulative probability and then convert it to the probability
of interest
 If we’re given a probability and need to find the value of
x , convert the probability to the related cumulative
probability, find the z-score using a normal table (or
software, or a calculator), and then evaluate x    z .
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