Transcript Lectures 12-13 - Rice University Statistics

The Mean of a Discrete
Probability Distribution

The mean of a probability distribution for a
discrete random variable is
   x  p(x)
where the sum is taken over all possible
values of x.
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Which Wager do You Prefer?

You are given $100 and told that you must pick
one of two wagers, for an outcome based on
flipping a coin:
A. You win $200 if it comes up heads and
lose $50 if it comes up tails.
B. You win $350 if it comes up head and lose
your original $100 if it comes up tails.

Without doing any calculation, which wager
would you prefer?
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You win $200 if it comes up heads and
lose $50 if it comes up tails.
Find the expected outcome for this
wager.
a.
b.
c.
d.
$100
$25
$50
$75
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You win $350 if it comes up head and lose
your original $100 if it comes up tails.
Find the expected outcome for this
wager.
a.
b.
c.
d.
$100
$125
$350
$275
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Section 6.2
How Can We Find Probabilities for
Bell-Shaped Distributions?
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Normal Distribution


The normal distribution is symmetric,
bell-shaped and characterized by its mean
µ and standard deviation σ.
The probability of falling within any
particular number of standard deviations
of µ is the same for all normal
distributions.
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Normal Distribution
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Z-Score

Recall: The z-score for an observation
is the number of standard deviations
that it falls from the mean.
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Z-Score

For each fixed number z, the probability
within z standard deviations of the mean
is the area under the normal curve
between
 - z and   z
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Z-Score

For z = 1:
68% of the area (probability) of a normal
distribution falls between:
 - 1 and   1
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Z-Score

For z = 2:
95% of the area (probability) of a normal
distribution falls between:
 - 2 and   2
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Z-Score

For z = 3:
Nearly 100% of the area (probability) of a normal
distribution falls between:
 - 3 and   3
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The Normal Distribution: The
Most Important One in Statistics

It’s important because…
• Many variables have approximate normal
•
•
distributions.
It’s used to approximate many discrete
distributions.
Many statistical methods use the normal
distribution even when the data are not
bell-shaped.
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Finding Normal Probabilities for
Various Z-values

Suppose we wish to find the
probability within, say, 1.43 standard
deviations of µ.
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Z-Scores and the Standard
Normal Distribution

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 have the standard normal
distribution.
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Example: Find the probability within
1.43 standard deviations of µ
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Example: Find the probability within
1.43 standard deviations of µ

Probability below 1.43σ = .9236

Probability above 1.43σ = .0764

By symmetry, probability below
-1.43σ = .0764

Total probability under the curve = 1
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Example: Find the probability within
1.43 standard deviations of µ
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Example: Find the probability within
1.43 standard deviations of µ

The probability falling within 1.43
standard deviations of the mean
equals:
1 – 0.1528 = 0.8472, about 85%
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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.
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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.
So, a probability of 0.025 lies below
µ - 1.96σ.
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Example: Find the Value of z For a
Cumulative Probability of 0.025
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Example: What IQ Do You Need
to Get Into Mensa?

Mensa is a society of high-IQ people
whose members have a score on an
IQ test at the 98th percentile or higher.
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Example: What IQ Do You Need
to Get Into Mensa?

How many standard deviations above
the mean is the 98th percentile?
• The cumulative probability of 0.980 in the
body of Table A corresponds to z = 2.05.
• The 98th percentile is 2.05 standard
deviations above µ.
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Example: What IQ Do You Need
to Get Into Mensa?

What is the IQ for that percentile?
• Since µ = 100 and σ 16, the 98th percentile
of IQ equals:
µ + 2.05σ = 100 + 2.05(16) = 133
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Z-Score for a Value of
a Random Variable

The z-score for a value of a random variable
is the number of standard deviations that x
falls from the mean µ.

It is calculated as:
z
x-

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Example: Finding Your Relative
Standing on The SAT

Scores on the verbal or math portion of
the SAT are approximately normally
distributed with mean µ = 500 and
standard deviation σ = 100. The scores
range from 200 to 800.
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Example: Finding Your Relative
Standing on The SAT

If one of your SAT scores was x = 650,
how many standard deviations from
the mean was it?
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Example: Finding Your Relative
Standing on The SAT

Find the z-score for x = 650.
x-
650 - 500
z

 1.50

100
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Example: Finding Your Relative
Standing on The SAT

What percentage of SAT scores was
higher than yours?
• Find the cumulative probability for the z•
score of 1.50 from Table A.
The cumulative probability is 0.9332.
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Example: Finding Your Relative
Standing on The SAT



The cumulative probability below 650
is 0.9332.
The probability above 650 is
1 – 0.9332 = 0.0668
About 6.7% of SAT scores are higher
than yours.
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Example: What Proportion of
Students Get A Grade of B?



On the midterm exam in introductory
statistics, an instructor always give a
grade of B to students who score between
80 and 90.
One year, the scores on the exam have
approximately a normal distribution with
mean 83 and standard deviation 5.
About what proportion of students
get a B?
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Example: What Proportion of
Students Get A Grade of B?

Calculate the z-score for 80 and for 90:
z
z
x-

x-

90 - 83

 1.40
5
80 - 83

 - 0.60
5
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Example: What Proportion of
Students Get A Grade of B?

Look up the cumulative probabilities in
Table A.
• For z = 1.40, cum. Prob. = 0.9192
• For z = -0.60, cum. Prob. = 0.2743

It follows that about 0.9192 – 0.2743 =
0.6449, or about 64% of the exam scores
were in the ‘B’ range.
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Using z-scores to Find
Normal Probabilities

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 to get a
cumulative probability.
Convert it to the probability of interest.
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Using z-scores to Find
Random Variable x Values

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.

Evaluate x = zσ + µ.
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Example: How Can We Compare Test
Scores That Use Different Scales?



When you applied to college, you scored 650
on an SAT exam, which had mean µ = 500
and standard deviation σ = 100.
Your friend took the comparable ACT in
2001, scoring 30. That year, the ACT had µ =
21.0 and σ = 4.7.
How can we tell who did better?
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What is the z-score for your SAT
score of 650?
For the SAT scores: µ = 500 and σ = 100.
a.
2.15
b. 1.50
c.
-1.75
d. -1.25
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What percentage of students
scored higher than you?
a.
b.
c.
d.
10%
5%
2%
7%
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What is the z-score for your
friend’s ACT score of 30?
The ACT scores had a mean of 21 and a
standard deviation of 4.7.
a.
1.84
b. -1.56
c.
1.91
d. -2.24
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What percentage of students
scored higher than your friend?
a.
b.
c.
d.
3%
6%
10%
1%
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Standard Normal Distribution


The standard normal distribution is
the normal distribution with mean
µ = 0 and standard deviation σ = 1.
It is the distribution of normal
z-scores.
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