agresti/franklin statistics, 1 of 139 learn …. to analyze how likely it is that sample results...
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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
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Inferential Statistics
Use sample data to make decisions and predictions about a population
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Section 6.1
How Can We Summarize Possible Outcomes and Their
Probabilities?
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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
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Random Variable
A random variable is a numerical measurement of the outcome of a random phenomenon.
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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.
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Probability Distribution
The probability distribution of a random variable specifies its possible values and their probabilities.
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Discrete Random Variable
The possible outcomes are a set of separate numbers: (0, 1,2, …).
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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
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Example: How many Home Runs Will the Red Sox Hit in a Game?
What is the estimated probability of at least three home runs?
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Example: How many Home Runs Will the Red Sox Hit in a Game?
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Parameters of a Probability Distribution
Parameters: numerical summaries of a probability distribution.
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The Mean of a Probability Distribution
The mean of a probability distribution is denoted by the parameter, µ.
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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
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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.
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Example: What’s the Expected Number of Home Runs in a Baseball Game?
Find the mean of this probability distribution.
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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
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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.
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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.
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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.
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Continuous Variables are Measured in a Discrete Manner because of Rounding.
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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. $100
b. $25
c. $50
d. $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. $100
b. $125
c. $350
d. $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.
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
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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:
-x
z
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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.
1.50 100
500 - 650
-x z
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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:
1.40 5
83 - 90
-x z
0.60- 5
83 - 80
-x z
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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:
Use a table of normal probabilities to get a cumulative probability.
Convert it to the probability of interest.
-x
z
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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. 10%
b. 5%
c. 2%
d. 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. 3%
b. 6%
c. 10%
d. 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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Section 6.3
How Can We Find Probabilities When Each Observation Has Two Possible
Outcomes?
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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.
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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.
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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.
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Example: Finding Binomial Probabilities for An ESP Experiment
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?
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Example: Finding Binomial Probabilities for An ESP Experiment
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Example: Finding Binomial Probabilities for An ESP Experiment
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.
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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) )
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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
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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.
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Example: Are Women Passed over for Managerial 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.
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Example: Are Women Passed over for Managerial Training?
How can we investigate statistically the women’s assertion of gender bias?
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Example: Are Women Passed over for Managerial Training?
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.
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Example: Are Women Passed over for Managerial Training?
Due to ordinary sampling variation, it need not happen that exactly 50 % of those selected are females.
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Example: Are Women Passed over for Managerial Training?
If employees were actually selected at random for the training, what are the chances that none of the 10 employees selected were females?
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Example: Are Women Passed over for Managerial Training?
The probability that no females are chosen equals:
001.0)50.0()50.0(0!10!
10! p(0) 100
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Example: Are Women Passed over for Managerial Training?
It is very unlikely (one chance in a thousand) that none of the 10 selected for management training would be female.
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Example: Are Women Passed over for Managerial Training?
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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.
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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.
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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,
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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.
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Example: How Can We Check for Racial Profiling?
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.
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Example: How Can We Check for Racial Profiling?
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)?
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Example: How Can We Check for Racial Profiling?
Assume:• 262 car stops represent n = 262 trials.
• Successive police car stops are independent.
• P(driver is African-American) is p = 0.422.
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Example: How Can We Check for Racial Profiling?
Calculate the mean and standard deviation of this binomial distribution:
p)-np(1 np,
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Example: How Can We Check for Racial Profiling?
8(0.578)262(0.422)
111 262(0.422)
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Example: How Can We Check for Racial Profiling?
Recall: Empirical Rule• When a distribution is bell-shaped, about
100% of it falls within 3 standard deviations of the mean.
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Example: How Can We Check for Racial Profiling?
1353(8)111 3
87 3(8) - 111 3 -u
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Example: How Can We Check for Racial Profiling?
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.
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Example: How Can We Check for Racial Profiling?
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.
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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.
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Section 6.4
How Likely Are the Possible Values of a Statistic?
The Sampling Distribution
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Statistic
Recall: A statistic is a numerical summary of sample data, such as a sample proportion or a sample mean.
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Parameter
Recall: A parameter is a numerical summary of a population, such as a population proportion or a population mean.
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Statistics and Parameters
In practice, we seldom know the values of parameters.
Parameters are estimated using sample data.
We use statistics to estimate parameters.
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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.
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Example: 2003 California Recall Election
If a different random sample of about 3000 voters were selected, a different sample proportion would occur.
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Example: 2003 California Recall Election
Imagine all the distinct samples of 3000 voters you could possibly get.
Each such sample has a value for the sample proportion.
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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.
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Sampling Distribution
The sampling distribution of a statistic is the probability distribution that specifies probabilities for the possible values the statistic can take.
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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.
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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.
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Example: Sampling Distribution
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
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Example: Sampling Distribution
Sample Proportion
Probability
0 1/8
1/3 3/8
2/3 3/8
1 1/8
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Example: Sampling Distribution
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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
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Example: 2003 California Recall Election
Sample: Exit poll of 3160 voters.
Suppose that exactly 50% of the population of all voters voted in favor of the recall.
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Example: 2003 California Recall Election
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
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Example: 2003 California Recall Election
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
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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.
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Example: 2003 California Recall Election
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?
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Example: 2003 California Recall Election
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.
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Example: 2003 California Recall Election
Convert the sample proportion value of 0.54 to a z-score:
4.5 0.0089
0.50) - (0.54 z
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Example: 2003 California Recall Election
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Example: 2003 California Recall Election
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.
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Example: 2003 California Recall Election
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.
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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
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Section 6.5
How Close Are Sample Means to Population Means?
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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.
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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
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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 = $300.
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Example: How Much Do Mean Sales Vary From Week to Week?
The mean sales for the seven days in a week are computed each week.
The weekly means are plotted over time. These weekly means form a sampling
distribution.
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Example: How Much Do Mean Sales Vary From Week to Week?
What are the center and spread of the sampling distribution?
113 7
300
900$
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Sampling Distribution vs. Population Distribution
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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.”
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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
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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?
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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.
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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.
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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.
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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.
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Section 6.6
How Can We Make Inferences About a Population?
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Three Distinct Types of Distributions
Population Distribution
Data Distribution
Sampling Distribution
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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.
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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.
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Sampling Distribution
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.