Agresti/Franklin Statistics, 1e, 1 of 139 The Mean of a Discrete Probability Distribution The mean...

Agresti/Franklin Statistics, 1e, 1 of 139

The Mean of a Discrete Probability Distribution

The mean of a probability distribution for a discrete random variable is

where the sum is taken over all possible values of x.

)(xpx


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?


You win $200 if it comes up heads and lose $50 if it comes up tails.

Find the expected outcome for this

wager.

a. $100

b. $25

c. $50

d. $75


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. $100

b. $125

c. $350

d. $275


Section 6.2

How Can We Find Probabilities for Bell-Shaped Distributions?


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.


Normal Distribution


Z-Score

Recall: The z-score for an observation is the number of standard deviations that it falls from the mean.


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 -


Z-Score

For z = 1:

68% of the area (probability) of a normal

distribution falls between:

1 and 1 -


Z-Score

For z = 2:

95% of the area (probability) of a normal


2 and 2 -


Z-Score

For z = 3:

Nearly 100% of the area (probability) of a normal


3 and 3 -


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.


Finding Normal Probabilities for Various Z-values

Suppose we wish to find the probability within, say, 1.43 standard deviations of µ.


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.


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





The probability falling within 1.43 standard deviations of the mean equals:

1 – 0.1528 = 0.8472, about 85%


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.


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.





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.



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.



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


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:

-x

z


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.



If one of your SAT scores was x = 650, how many standard deviations from the mean was it?



Find the z-score for x = 650.

1.50 100

500 - 650

-x z



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.



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.


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?



Calculate the z-score for 80 and for 90:

1.40 5

83 - 90

-x z

0.60- 5

83 - 80

-x z



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.


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:

Use a table of normal probabilities to get a cumulative probability.

Convert it to the probability of interest.

-x

z


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.


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?


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


What percentage of students scored higher than you?

a. 10%

b. 5%

c. 2%

d. 7%


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


What percentage of students scored higher than your friend?

a. 3%

b. 6%

c. 10%

d. 1%


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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