Chapter 6 probability distributions
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Chapter 6: Probability Distributions. Section 6.1: How Can We Summarize Possible Outcomes and Their Probabilities?. Learning Objectives. Random variable Probability distributions for discrete random variables Mean of a probability distribution

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Chapter 6: Probability Distributions

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Chapter 6 probability distributions

Chapter 6: Probability Distributions

Section 6.1: How Can We Summarize Possible Outcomes and Their Probabilities?


Learning objectives

Learning Objectives

  • Random variable

  • Probability distributions for discrete random variables

  • Mean of a probability distribution

  • Summarizing the spread of a probability distribution

  • Probability distribution for continuous random variables


Learning objective 1 randomness

Learning Objective 1: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


Learning objective 1 random variable

Learning Objective 1:Random Variable

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


Learning objective 1 random variable1

Learning Objective 1:Random Variable

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

    • Variables

    • A particular value of the random variable

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

    Example: Flip a coin three times

    • X=number of heads in the 3 flips; defines the random variable

    • x=2; represents a possible value of the random variable


Learning objective 2 probability distribution

Learning Objective 2:Probability Distribution

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

    Note: It is the randomness of the variable that allows us to specify probabilities for the outcomes


Learning objective 2 probability distribution of a discrete random variable

Learning Objective 2:Probability Distribution of a Discrete Random Variable

  • A discrete random variableX has separate values (such as 0,1,2,…) as its possible outcomes

  • 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


Learning objective 2 example

Learning Objective 2:Example

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

    P(3)+P(4)+P(5)=0.13+0.03+0.01=0.17


Learning objective 3 the mean of a discrete probability distribution

Learning Objective 3: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.

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

  • The mean is a weighted average; values of x that are more likely receive greater weight P(x)


Learning objective 3 expected value of x

Learning Objective 3: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.

  • It is not unusual for the expected value of a random variable to equal a number that is NOT a possible outcome.


Learning objective 3 example

Learning Objective 3:Example

  • Find the mean of this probability distribution.

The mean:

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


Learning objective 4 the standard deviation of a probability distribution

Learning Objective 4: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.

  • Roughly, σ describes how far the random variable falls, on the average, from the mean of its distribution


Learning objective 5 continuous random variable

Learning Objective 5: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 measured in a discrete manner because of rounding.


Learning objective 5 probability distribution of a continuous random variable

Learning Objective 5: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.

  • Each interval has probability between 0 and 1.

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


Chapter 6 probability distributions1

Chapter 6: Probability Distributions

Section 6.2: How Can We Find Probabilities for Bell-Shaped Distributions?


Learning objectives1

Learning Objectives

  • Normal Distribution

  • 68-95-99.7 Rule for normal distributions

  • Z-Scores and the Standard Normal Distribution

  • The Standard Normal Table: Finding Probabilities

  • Using the TI-calculator: find probabilities


Learning objectives2

Learning Objectives

  • Using the Standard Normal Table in Reverse

  • Using the TI-calculator: find z-scores

  • Probabilities for Normally Distributed Random Variables

  • Percentiles for Normally Distributed Random Variables

  • Using Z-scores to Compare Distributions


Learning objective 1 normal distribution

Learning Objective 1:Normal Distribution

The normal distribution is symmetric, bell-shaped and characterized by its mean µ and standard deviation .

  • The normal distribution is the most important distribution in statistics

    • Many distributions have an approximate normal distribution

    • Approximates 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


Learning objective 1 normal distribution1

Learning Objective 1: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


Learning objective 1 normal distribution2

Learning Objective 1:Normal Distribution

  • Within what interval do almost all of the men’s heights fall? Women’s height?


Learning objective 2 68 95 99 7 rule for any normal curve

Learning Objective 2:68-95-99.7 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


Learning objective 2 example 68 95 99 7 rule

Learning Objective 2: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

      [ µ = 65  3.5 ]

    • 95% are between 58 and 72 inches

      [ µ 2 = 65  2(3.5) = 65  7 ]

    • 99.7% are between 54.5 and 75.5 inches

      [ µ 3 = 65  3(3.5) = 65  10.5 ]


Learning objective 2 example 68 95 99 7 rule1

68%

(by 68-95-99.7 Rule)

?

16%

-1

+1

65 68.5 (height values)

? = 84%

Learning Objective 2:Example : 68-95-99.7% Rule

  • What proportion of women are less than 69 inches tall?


Learning objective 3 z scores and the standard normal distribution

Learning Objective 3: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

  • 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


Learning objective 3 z scores and the standard normal distribution1

Learning Objective 3:Z-Scores and the Standard Normal Distribution

  • A standard normal distribution has mean µ=0 and standard deviation σ=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 have the standard normal distribution.


Learning objective 4 table a standard normal probabilities

Learning Objective 4:Table A: Standard Normal Probabilities

Table A 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


Learning objective 4 example using table a

Learning Objective 4:Example: Using Table A

  • Find the probability that a normal random variable takes a value less than 1.43 standard deviations above µ; P(z<1.43)=.9236

TI Calculator = Normcdf(-1e99,1.43,0,1)= .9236


Learning objective 4 example using table a1

Learning Objective 4:Example: Using Table A

  • Find the probability that a normal random variable takes a value greater than 1.43 standard deviations above µ: P(z>1.43)=1-.9236=.0764

TI Calculator = Normcdf(1.43,1e99,0,1)= 0.0764


Learning objective 4 example

Learning Objective 4:Example:

  • Find the probability that a normal random variable assumes a value within 1.43 standard deviations of µ

    • Probability below 1.43σ = .9236

    • Probability below -1.43σ = .0764 (1-.9236)

    • P(-1.43<z<1.43) =.9236-.0764=.8472

TI Calculator = Normcdf(-1.43,1.43,0,1)= .8472


Learning objective 5 using the ti calculator

Learning Objective 5:Using the TI Calculator

To calculate the cumulative probability

  • 2nd DISTR; 2:normalcdf(lower bound, upper bound,mean,sd)

  • Use –1E99 for negative infinity and 1E99 for positive infinity


Learning objective 5 find probabilities using ti calculator

Learning Objective 5:Find Probabilities Using TI Calculator

  • Find probability to the left of -1.64

    • P(z<-1.64)=normcdf(-1e99,-1.64,0,1)=.0505

  • Find probability to the right of 1.56

    • P(z>1.56)=normcdf(1.56,1e99,0,1)=.0594

  • Find probability between -.50 and 2.25

    • P(-.5<z<2.25)=normcdf(-.5,2.25,0,1)=.6793


Learning objective 6 how can we find the value of z for a certain cumulative probability

Learning Objective 6: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


Learning objective 6 how can we find the value of z for a certain cumulative probability1

Learning Objective 6: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.


Learning objective 6 how can we find the value of z for a certain cumulative probability2

Learning Objective 6:How Can We Find the Value of z for a Certain Cumulative Probability?

  • Example: Find the value of z for a cumulative probability of 0.975.

  • Look up the cumulative probability of 0.975 in the body of Table A.

  • A cumulative probability of 0.975 corresponds to z = 1.96.

  • Thus, the probability that a normal

    random variable takes a value no more

    than 1.96 standard deviations above

    the mean is 0.975.


Learning objective 7 using the ti calculator to find z scores for a given probability

Learning Objective 7:Using the TI Calculator to Find Z-Scores for a Given Probability

  • 2nd DISTR 3:invNorm; Enter

  • invNorm(percentile,mean,sd)

    • Percentile is the probability under the curve from negative infinity to the z-score

  • Enter


Learning objective 7 examples

Learning Objective 7:Examples

  • The probability that a standard normal random variable assumes a value that is ≤ z is 0.975. What is z? Invnorm(.975,0,1)=1.96

  • The probability that a standard normal random variable assumes a value that is > z is 0.0275.

    What is z? Invnorm(.975,0,1)=1.96

  • The probability that a standard normal random variable assumes a value that is ≥ z is 0.881.

    What is z? Invnorm(1-.881,0,1)=-1.18

  • The probability that a standard normal random variable assumes a value that is < z is 0.119.

    What is z? Invnorm(.119,0,1)= -1.18


Learning objective 7 example

Learning Objective 7:Example

  • Find the z-score z such that the probability within z standard deviations of the mean is 0.50.

    • Invnorm(.75,0,1)= .67

    • Invnorm(.25,0,1)= -.67

  • Probability = P(-.67<Z<.67)=.5


Learning objective 8 finding probabilities for normally distributed random variables

Learning Objective 8:Finding Probabilities for Normally Distributed Random Variables

  • State the problem in terms of the observed random variable X, i.e., P(X<x)

  • Standardize X to restate the problem in terms of a standard normal variable Z

  • Draw a picture to show the desired probability under the standard normal curve

  • Find the area under the standard normal curve using Table A


Learning objective 8 p x x

Learning Objective 8:P(X<x)

  • Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure less than 100?

  • P(X<100) =

  • Normcdf(-1E99,100,120,20)=.1587

  • 15.9% of adults have systolic blood pressure less than 100


Learning objective 8 p x x1

Learning Objective 8:P(X>x)

  • Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure greater than 100?

  • P(X>100) = 1 – P(X<100)

  • P(X>100)= 1-.1587=.8413

  • Normcdf(100,1e99,120,20)=.8413

  • 84.1% of adults have systolic blood pressure greater than 100


Learning objective 8 p x x2

Learning Objective 8:P(X>x)

  • Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure greater than 133?

  • P(X>133) = 1 – P(X<133)

  • P(X>133)= 1-.7422=.2578

  • Normcdf(133,1E99,120,20)=.2578

  • 25.8% of adults have systolic blood pressure greater than 133


Learning objective 8 p a x b

Learning Objective 8: P(a<X<b)

  • Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What percentage of adults have systolic blood pressure between 100 and 133?

  • P(100<X<133) = P(X<133)-P(X<100)

  • Normcdf(100,133,120,20)=.5835

  • 58% of adults have systolic blood pressure between 100 and 133


Learning objective 9 find x value given area to left

Learning Objective 9:Find X Value Given Area to Left

  • Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. What is the 1st quartile?

  • P(X<x)=.25, find x:

    • Look up .25 in the body of Table A to find z= -0.67

    • Solve equation to find x:

  • Check:

    • P(X<106.6) P(Z<-0.67)=0.25

    • TI Calculator = Invnorm(.25,120,20)=106.6


Learning objective 9 find x value given area to right

Learning Objective 9:Find X Value Given Area to Right

  • Adult systolic blood pressure is normally distributed with µ = 120 and σ = 20. 10% of adults have systolic blood pressure above what level?

  • P(X>x)=.10, find x.

    • P(X>x)=1-P(X<x)

    • Look up 1-0.1=0.9 in the body of Table A to find z=1.28

    • Solve equation to find x:

  • Check:

    • P(X>145.6) =P(Z>1.28)=0.10

    • TI Calculator = Invnorm(.9,120,20)=145.6


Learning objective 10 using z scores to compare distributions

Learning Objective 10:Using Z-scores to Compare Distributions

Z-scores can be used to compare observations from different normal distributions

  • Example:

    • You score 650 on the SAT which has =500 and

      =100 and 30 on the ACT which has =21.0 and

      =4.7. On which test did you perform better?

    • Compare z-scores

      SAT: ACT:

    • Since your z-score is greater for the ACT, you performed better on this exam


Chapter 6 probability distributions2

Chapter 6: Probability Distributions

Section 6.3: How Can We Find Probabilities When Each Observation Has Two Possible Outcomes?


Learning objectives3

Learning Objectives

  • The Binomial Distribution

  • Conditions for a Binomial Distribution

  • Probabilities for a Binomial Distribution

  • Factorials

  • Examples using Binomial Distribution

  • Do the Binomial Conditions Apply?

  • Mean and Standard Deviation of the Binomial Distribution

  • Normal Approximation to the Binomial


Learning objective 1 the binomial distribution

Learning Objective 1: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 do not have, health insurance.

    • Vote yes or no on a referendum.


Learning objective 2 conditions for the binomial distribution

Learning Objective 2:Conditions for the Binomial Distribution

  • Each of n trials has two possible outcomes: “success” or “failure”.

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

  • The ntrials are independent.

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


Learning objective 3 probabilities for a binomial distribution

Learning Objective 3: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:


Learning objective 4 factorials

Learning Objective 4:Factorials

Rules for factorials:

  • n!=n*(n-1)*(n-2)…2*1

  • 1!=1

  • 0!=1

    For example,

  • 4!=4*3*2*1=24


Learning objective 5 example finding binomial probabilities

Learning Objective 5:Example: Finding Binomial Probabilities

  • 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.

    • Three trials are performed for the experiment.

    • Doe got the correct answer twice.


Learning objective 5 example 1

Learning Objective 5:Example 1

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(0.032)=0.096.


Learning objective 5 example 11

Learning Objective 5:Example 1

  • 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.

2nd Vars

0:binampdf(n,p,x)

Binampdf(3,.2,2)=0.096


Learning objective 5 binomial example 2

Learning Objective 5:Binomial Example 2

  • 1000 employees, 50% Female

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

  • The probability that no females are chosen is:

  • Binompdf(10,.5,0)=9.765625E-4

  • It is very unlikely (one chance in a thousand) that none of the 10 selected for management training would be female if the employees were chosen randomly


Learning objective 6 do the binomial conditions apply

Learning Objective 6:Do the Binomial Conditions Apply?

  • Before using 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.


Learning objective 6 do the binomial conditions apply to example 2

Learning Objective 6:Do the Binomial Conditions Apply to Example 2?

  • The data are binary (male, female).

  • If employees are selected randomly, the probability of selecting a female on a given trial is 0.50.

  • With random sampling of 10 employees from a large population, outcomes for one trial does not depend on the outcome of another trial


Learning objective 7 binomial mean and standard deviation

Learning Objective 7: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:


Learning objective 7 example racial profiling

Learning Objective 7: Example: 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.

    • 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)?


Learning objective 7 example racial profiling1

Learning Objective 7:Example: 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.

  • Calculate the mean and standard deviation of this binomial distribution:


Learning objective 7 example racial profiling2

Learning Objective 7: Example: Racial Profiling?

  • Recall: Empirical Rule

    • When a distribution is bell-shaped, close to 100% of the observations fall within 3 standard deviations of the mean.


Learning objective 7 example racial profiling3

Learning Objective 7:Example: Racial Profiling?

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

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

  • The number of African-Americans 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.


Learning objective 8 approximating the binomial distribution with the normal distribution

Learning Objective 8:Approximating the Binomial Distribution with the Normal Distribution

  • The binomial distribution can be well approximated by the normal distribution 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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