Agresti/Franklin Statistics, 1 of 139 Learn …. To analyze how likely it is that sample results...

Agresti/Franklin Statistics, 1 of 139

Learn ….To analyze how likely it is that sample results will be “close” to population values

How probability provides the basis for making statistical inferences

Chapter 6Probability Distributions

Agresti/Franklin Statistics, 1e, 2 of 139

Inferential Statistics

Use sample data to make decisions and predictions about a population


Section 6.1

How Can We Summarize Possible Outcomes and Their

Probabilities?


Randomness

The numerical values that a variable assumes are the result of some random phenomenon:• Selecting a random sample for a

population

or

• Performing a randomized experiment


Random Variable

A random variable is a numerical measurement of the outcome of a random phenomenon.


Random Variable

Use letters near the end of the alphabet, such as x, to symbolize variables.

Use a capital letter, such as X, to refer to the random variable itself.

Use a small letter, such as x, to refer to a particular value of the variable.


Probability Distribution

The probability distribution of a random variable specifies its possible values and their probabilities.


Discrete Random Variable

The possible outcomes are a set of separate numbers: (0, 1,2, …).


Probability Distribution of a Discrete Random Variable

A discrete random variable X takes a set of separate values (such as 0,1,2,…)

Its probability distribution assigns a probability P(x) to each possible value x:

• For each x, the probability P(x) falls between 0 and 1

• The sum of the probabilities for all the possible x values equals 1


Example: How many Home Runs Will the Red Sox Hit in a Game?

What is the estimated probability of at least three home runs?


Example: How many Home Runs Will the Red Sox Hit in a Game?


Parameters of a Probability Distribution

Parameters: numerical summaries of a probability distribution.


The Mean of a Probability Distribution

The mean of a probability distribution is denoted by the parameter, µ.


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


Expected Value of X

The mean of a probability distribution of a random variable X is also called the expected value of X.

The expected value reflects not what we’ll observe in a single observation, but rather that we expect for the average in a long run of observations.


Example: What’s the Expected Number of Home Runs in a Baseball Game?

Find the mean of this probability distribution.


Example: What’s the Expected Number of Home Runs in a Baseball Game?

The mean:

= 0(0.23) + 1(0.38) + 2(0.22) + 3(0.13) + 4(0.03) + 5(0.01) = 1.38

)(xpx


The Standard Deviation of a Probability Distribution

The standard deviation of a probability distribution, denoted by the parameter, σ, measures its spread.

Larger values of σ correspond to greater spread.


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.


Probability Distribution of a Continuous Random Variable

A continuous random variable has possible values that from an interval.

Its probability distribution is specified by a curve.

Each interval has probability between 0 and 1.

The interval containing all possible values has probability equal to 1.


Continuous Variables are Measured in a Discrete Manner because of Rounding.


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.


Section 6.3

How Can We Find Probabilities When Each Observation Has Two Possible

Outcomes?


The Binomial Distribution

Each observation is binary: it has one of two possible outcomes.

Examples:• Accept, or decline an offer from a bank for a credit

card.

• Have, or have not, health insurance.

• Vote yes or no in a referendum.


Conditions for the Binomial Distribution

Each of n trails has two possible outcomes: “success” and “failure”.

Each trail has the same probability of success, denoted by p.

The n trials are independent.

The binomial random variable X is the number of successes in the n trials.


Example: Finding Binomial Probabilities for An ESP Experiment

John Doe claims to possess ESP. An experiment is conducted:

• A person in one room picks one of the integers 1, 2, 3, 4, 5 at random.

• In another room, John Doe identifies the number he believes was picked.

• The experiment is done with three trials.

• Doe got the correct answer twice.



If John Doe does not actually have ESP and is actually guessing the number, what is the probability that he’d make a correct guess on two of the three trials?



The three ways John Doe could make two correct guesses in three trials are: SSF, SFS, and FSS.

Each of these has probability: (0.2)2(0.8)=0.032.

The total probability of two correct guesses is 3(0.2)2(0.8)=0.096.


Probabilities for a Binomial Distribution

Denote the probability of success on a trial by p.

For n independent trials, the probability of x successes equals:

nxpp xnx ,...,2,1,0,)1(x)!-(nx!

n!p(x) )


Example: Using the Binomial Formula in ESP Experiment

The probability of exactly 2 correct guesses is the binomial probability with n = 3 trials, x = 2 correct guesses and p = 0.2 probability of a correct guess.

0.096 8)3(0.04)(0. )8.0()2.0(2!1!

3! p(2) 12


Example: Are Women Passed over for Managerial Training?

Example: Presence of bias in promotion.• Large supermarket in Florida.

• Group of women claimed that female employees were passed over for management training.



Large employee pool of more than 1000 people.

Half the employees are male; half are female.

None of the 10 employees chosen for management training were female.



How can we investigate statistically the women’s assertion of gender bias?



If the employees are selected randomly in terms of gender, about half of the employees picked should be females and about half should be males.



Due to ordinary sampling variation, it need not happen that exactly 50 % of those selected are females.



If employees were actually selected at random for the training, what are the chances that none of the 10 employees selected were females?



The probability that no females are chosen equals:

001.0)50.0()50.0(0!10!

10! p(0) 100



It is very unlikely (one chance in a thousand) that none of the 10 selected for management training would be female.


Do the Binomial Conditions Apply?

Before you use the binomial distribution, check that its three conditions apply:

• Binary data (success or failure).

• The same probability of success for each trial (denoted by p).

• Independent trials.


Do the Binomial Conditions Apply to the Managerial Training Example?

The data are binary. If employees are selected randomly, the

probability of selecting a female on a given trial is 0.50.

With random sampling from a large population, outcomes from trials are independent.


Binomial Mean and Standard Deviation

The binomial probability distribution for n trials with probability p of success on each trial has mean µ and standard deviation σ given by:

p)-np(1 np,


Example: How Can We Check for Racial Profiling?

Study conducted by the American Civil Liberties Union.

Study analyzed whether African-American drivers were more likely than other in the population to be targeted by police for traffic stops.



Data:• 262 police car stops in Philadelphia in

1997.

• 207 of the drivers stopped were African-American.

• In 1997, Philadelphia’s population was 42.2% African-American.



Does the number of African-Americans stopped suggest possible bias, being higher than we would expect (other things being equal, such as the rate of violating traffic laws)?



Assume:• 262 car stops represent n = 262 trials.

• Successive police car stops are independent.

• P(driver is African-American) is p = 0.422.



Calculate the mean and standard deviation of this binomial distribution:

p)-np(1 np,



8(0.578)262(0.422)

111 262(0.422)



Recall: Empirical Rule• When a distribution is bell-shaped, about

100% of it falls within 3 standard deviations of the mean.



1353(8)111 3

87 3(8) - 111 3 -u



If no racial profiling is happening, we would not be surprised if between about 87 and 135 of the 262 people stopped were African-American.

The actual number stopped (207) is well above these values.

The number of African-American stopped is too high, even taking into account random variation.



Limitation of the analysis:• Different people do different

amounts of driving, so we don’t really know that 42.2% of the potential stops were African-American.


When Is the Binomial Distribution Dell Shaped?

The binomial distribution has close to a symmetric, bell shape when the expected number of successes, np, and the expected number of failures, n(1-p) are both at least 15.


Section 6.4

How Likely Are the Possible Values of a Statistic?

The Sampling Distribution


Statistic

Recall: A statistic is a numerical summary of sample data, such as a sample proportion or a sample mean.


Parameter

Recall: A parameter is a numerical summary of a population, such as a population proportion or a population mean.


Statistics and Parameters

In practice, we seldom know the values of parameters.

Parameters are estimated using sample data.

We use statistics to estimate parameters.


Example: 2003 California Recall Election

Prior to counting the votes, the proportion in favor of recalling Governor Gray Davis was an unknown parameter.

An exit poll of 3160 voters reported that the sample proportion in favor of a recall was 0.54.



If a different random sample of about 3000 voters were selected, a different sample proportion would occur.



Imagine all the distinct samples of 3000 voters you could possibly get.

Each such sample has a value for the sample proportion.


Statistics and Parameters

How do we know that a sample statistic is a good estimate of a population parameter?

To answer this, we need to look at a probability distribution called the sampling distribution.


Sampling Distribution

The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take.


The Sampling Distribution of the Sample Proportion

Look at each possible sample. Find the sample proportion for each

sample. Construct the frequency distribution of

the sample proportion values. This frequency distribution is the

sampling distribution of the sample proportion.


Example: Sampling Distribution

Which Brand of Pizza Do You Prefer?

• Two Choices: A or D.

• Assume that half of the population prefers Brand A and half prefers Random D.

• Take a random sample of n = 3 tasters.



Sample No. Prefer Pizza A

Proportion

(A,A,A) 3 1

(A,A,D) 2 2/3

(A,D,A) 2 2/3

(D,A,A) 2 2/3

(A,D,D) 1 1/3

(D,A,D) 1 1/3

(D,D,A) 1 1/3

(D,D,D) 0 0



Sample Proportion

Probability

0 1/8

1/3 3/8

2/3 3/8

1 1/8


Mean and Standard Deviation of the Sampling Distribution of a Proportion

For a binomial random variable with n trials and probability p of success for each, the sampling distribution of the proportion of successes has:

To obtain these value, take the mean np and standard deviation for the binomial distribution of the number of successes and divide by n.

n

p)-p(1deviation standard and pMean

)1( pnp



Sample: Exit poll of 3160 voters.

Suppose that exactly 50% of the population of all voters voted in favor of the recall.



Describe the mean and standard deviation of the sampling distribution of the number in the sample who voted in favor of the recall.

• µ = np = 3160(0.50) = 1580

• 1.28)50.0)(50.0(3160p)-np(1



Describe the mean and standard deviation of the sampling distribution of the proportion in the sample who voted in favor of the recall.

0089.0000079.03160

)50.0)(50.0()1(Deviation Standard

0.50 p Mean

p

pp


The Standard Error

To distinguish the standard deviation of a sampling distribution from the standard deviation of an ordinary probability distribution, we refer to it as a standard error.



If the population proportion supporting recall was 0.50, would it have been unlikely to observe the exit-poll sample proportion of 0.54?

Based on your answer, would you be willing to predict that Davis would be recalled from office?



Fact: The sampling distribution of the sample proportion has a bell-shape with a mean µ = 0.50 and a standard deviation σ = 0.0089.



Convert the sample proportion value of 0.54 to a z-score:

4.5 0.0089

0.50) - (0.54 z



The sample proportion of 0.54 is more than four standard errors from the expected value of 0.50.

The sample proportion of 0.54 voting for recall would be very unlikely if the population support were p = 0.50.



A sample proportion of 0.54 would be even more unlikely if the population support were less than 0.50.

We there have strong evidence that the population support was larger than 0.50.

The exit poll gives strong evidence that Governor Davis would be recalled.


Summary of the Sampling Distribution of a Proportion

For a random sample of size n from a population with proportion p, the sampling distribution of the sample proportion has

If n is sufficiently large such that the expected numbers of outcomes of the two types, np and n(1-p), are both at least 15, then this sampling distribution has a bell-shape.

n

p)-p(1error standard and pMean


Section 6.5

How Close Are Sample Means to Population Means?


The Sampling Distribution of the Sample Mean

The sample mean, x, is a random variable.

The sample mean varies from sample to sample.

By contrast, the population mean, µ, is a single fixed number.


Mean and Standard Error of the Sampling Distribution of the Sample Mean

For a random sample of size n from a population having mean µ and standard deviation σ, the sampling distribution of the sample mean has:

• Center described by the mean µ (the same as the mean of the population).

• Spread described by the standard error, which equals the population standard deviation divided by the square root of the sample size: n


Example: How Much Do Mean Sales Vary From Week to Week?

Daily sales at a pizza restaurant vary from day to day.

The sales figures fluctuate around a mean µ = $900 with a standard deviation σ = $300.



The mean sales for the seven days in a week are computed each week.

The weekly means are plotted over time. These weekly means form a sampling

distribution.



What are the center and spread of the sampling distribution?

113 7

300

900$


Sampling Distribution vs. Population Distribution


Standard Error

Knowing how to find a standard error gives us a mechanism for understanding how much variability to expect in sample statistics “just by chance.”


Standard Error The standard error of the sample mean:

As the sample size n increases, the denominator increase, so the standard error decreases.

With larger samples, the sample mean is more likely to fall close to the population mean.

n


Central Limit Theorem

Question: How does the sampling distribution of the sample mean relate with respect to shape, center, and spread to the probability distribution from which the samples were taken?


Central Limit Theorem

For random sampling with a large sample size n, the sampling distribution of the sample mean is approximately a normal distribution.

This result applies no matter what the shape of the probability distribution from which the samples are taken.


Central Limit Theorem: How Large a Sample?

The sampling distribution of the sample mean takes more of a bell shape as the random sample size n increases. The more skewed the population distribution, the larger n must be before the shape of the sampling distribution is close to normal. In practice, the sampling distribution is usually close to normal when the sample size n is at least about 30.


A Normal Population Distribution and the Sampling Distribution

If the population distribution is approximately normal, then the sampling distribution is approximately normal for all sample sizes.


How Does the Central Limit Theorem Help Us Make Inferences

For large n, the sampling distribution is approximately normal even if the population distribution is not.

This enables us to make inferences about population means regardless of the shape of the population distribution.


Section 6.6

How Can We Make Inferences About a Population?


Three Distinct Types of Distributions

Population Distribution

Data Distribution



Population Distribution

This is the probability distribution from which we take the sample.

Value of its parameters, such as the population proportion p and the population mean µ are usually unknown.


Data Distribution This is the distribution of the sample data.

It’s described by sample statistics, such as a sample proportion or a sample mean.

With random sampling, the large the sample size n, the more closely it resembles the population distribution.

With larger n, the higher the probability that a sample statistic falls close to the population parameter.



This is the probability distribution of a sample statistic, such as a sample proportion or sample mean.

It provides the key for telling us how close a sample statistic falls to the unknown parameter.

For large n, the sampling distribution is approximately a normal distribution.

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