Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 1 of 34 Chapter 11 Section 1 Random Variables

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 2 of 34 Chapter 11 – Section 1 ●Learning objectives  Distinguish between discrete and continuous random variables  Identify discrete probability distributions  Construct probability histograms  Compute and interpret the mean of a discrete random variable  Interpret the mean of a discrete random variable as an expected value  Compute the variance and standard deviation of a discrete random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 3 of 34 Chapter 11 – Section 1 ●Learning objectives  Distinguish between discrete and continuous random variables  Identify discrete probability distributions  Construct probability histograms  Compute and interpret the mean of a discrete random variable  Interpret the mean of a discrete random variable as an expected value  Compute the variance and standard deviation of a discrete random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 4 of 34 Chapter 11 – Section 1 ●A random variable is a numeric measure of the outcome of a probability experiment  Random variables reflect measurements that can change as the experiment is repeated  Random variables are denoted with capital letters, typically using X (and Y and Z …)  Values are usually written with lower case letters, typically using x (and y and z...)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 5 of 34 Chapter 11 – Section 1 ●Examples ●Tossing four coins and counting the number of heads  The number could be 0, 1, 2, 3, or 4  The number could change when we toss another four coins ●Examples ●Tossing four coins and counting the number of heads  The number could be 0, 1, 2, 3, or 4  The number could change when we toss another four coins ●Measuring the heights of students  The heights could change from student to student

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 6 of 34 Chapter 11 – Section 1 ●A discrete random variable is a random variable that has either a finite or a countable number of values  A finite number of values such as {0, 1, 2, 3, and 4}  A countable number of values such as {1, 2, 3, …} ●A discrete random variable is a random variable that has either a finite or a countable number of values  A finite number of values such as {0, 1, 2, 3, and 4}  A countable number of values such as {1, 2, 3, …} ●Discrete random variables are designed to model discrete variables ●Discrete random variables are often “counts of …”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 7 of 34 Chapter 11 – Section 1 ●An example of a discrete random variable ●The number of heads in tossing 3 coins (a finite number of possible values) ●An example of a discrete random variable ●The number of heads in tossing 3 coins (a finite number of possible values)  There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads ●An example of a discrete random variable ●The number of heads in tossing 3 coins (a finite number of possible values)  There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads  A finite number of possible values – a discrete random variable ●An example of a discrete random variable ●The number of heads in tossing 3 coins (a finite number of possible values)  There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads  A finite number of possible values – a discrete random variable  This fits our general concept that discrete random variables are often “counts of …”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 8 of 34 Chapter 11 – Section 1 ●Other examples of discrete random variables ●The possible rolls when rolling a pair of dice  A finite number of possible pairs, ranging from (1,1) to (6,6) ●Other examples of discrete random variables ●The possible rolls when rolling a pair of dice  A finite number of possible pairs, ranging from (1,1) to (6,6) ●The number of pages in statistics textbooks  A countable number of possible values ●Other examples of discrete random variables ●The possible rolls when rolling a pair of dice  A finite number of possible pairs, ranging from (1,1) to (6,6) ●The number of pages in statistics textbooks  A countable number of possible values ●The number of visitors to the White House in a day  A countable number of possible values

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 9 of 34 Chapter 11 – Section 1 ●A continuous random variable is a random variable that has an infinite, and more than countable, number of values  The values are any number in an interval ●A continuous random variable is a random variable that has an infinite, and more than countable, number of values  The values are any number in an interval ●Continuous random variables are designed to model continuous variables (see section 1.1) ●Continuous random variables are often “measurements of …”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 10 of 34 Chapter 11 – Section 1 ●An example of a continuous random variable ●The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit ●An example of a continuous random variable ●The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit  The possible values (assuming that we can measure temperature to great accuracy) are in an interval ●An example of a continuous random variable ●The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit  The possible values (assuming that we can measure temperature to great accuracy) are in an interval  The interval may be something like (–20,110) ●An example of a continuous random variable ●The possible temperature in Chicago at noon tomorrow, measured in degrees  The possible values (assuming that we can measure temperature to great accuracy) are in an interval  The interval may be something like (–20,110)  This fits our general concept that continuous random variables are often “measurements of …”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 11 of 34 Chapter 11 – Section 1 ●Other examples of continuous random variables ●The height of a college student  A value in an interval between 3 and 8 feet ●Other examples of continuous random variables ●The height of a college student  A value in an interval between 3 and 8 feet ●The length of a country and western song  A value in an interval between 1 and 15 minutes ●Other examples of continuous random variables ●The height of a college student  A value in an interval between 3 and 8 feet ●The length of a country and western song  A value in an interval between 1 and 15 minutes

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 12 of 34 Chapter 11 – Section 1 ●Learning objectives  Distinguish between discrete and continuous random variables  Identify discrete probability distributions  Construct probability histograms  Compute and interpret the mean of a discrete random variable  Interpret the mean of a discrete random variable as an expected value  Compute the variance and standard deviation of a discrete random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 13 of 34 Chapter 11 – Section 1 ●The probability distribution of a discrete random variable X relates the values of X with their corresponding probabilities ●A distribution could be  In the form of a table  In the form of a graph  In the form of a mathematical formula

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 14 of 34 Chapter 11 – Section 1 ●If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x ●Examples  In tossing one coin, if X is the number of heads, then P(0) = 0.5 and P(1) = 0.5  In rolling one die, if X is the number rolled, then P(1) = 1/6

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 15 of 34 Chapter 11 – Section 1 ●Properties of P(x) ●Since P(x) form a probability distribution, they must satisfy the rules of probability  0 ≤ P(x) ≤ 1  Σ P(x) = 1 ●In the second rule, the Σ sign means to add up the P(x)’s for all the possible x’s

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 16 of 34 Chapter 11 – Section 1 ●An example of a discrete probability distribution ●All of the P(x) values are positive and they add up to 1 xP(x)P(x)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 17 of 34 Chapter 11 – Section 1 ●An example that is not a probability distribution ●Two things are wrong xP(x)P(x) ●An example that is not a probability distribution ●Two things are wrong  P(5) is negative xP(x)P(x) ●An example that is not a probability distribution ●Two things are wrong  P(5) is negative  The P(x)’s do not add up to 1 xP(x)P(x)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 18 of 34 Chapter 11 – Section 1 ●Learning objectives  Distinguish between discrete and continuous random variables  Identify discrete probability distributions  Construct probability histograms  Compute and interpret the mean of a discrete random variable  Interpret the mean of a discrete random variable as an expected value  Compute the variance and standard deviation of a discrete random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 19 of 34 Chapter 11 – Section 1 ●A probability histogram is a histogram where  The horizontal axis corresponds to the possible values of X (i.e. the x’s)  The vertical axis corresponds to the probabilities for those values (i.e. the P(x)’s) ●A probability histogram is very similar to a relative frequency histogram

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 20 of 34 Chapter 11 – Section 1 ●An example of a probability histogram ●The histogram is drawn so that the height of the bar is the probability of that value

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 21 of 34 Chapter 11 – Section 1 ●Learning objectives  Distinguish between discrete and continuous random variables  Identify discrete probability distributions  Construct probability histograms  Compute and interpret the mean of a discrete random variable  Interpret the mean of a discrete random variable as an expected value  Compute the variance and standard deviation of a discrete random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 22 of 34 Chapter 11 – Section 1 ●The mean of a probability distribution can be thought of in this way:  There are various possible values of a discrete random variable ●The mean of a probability distribution can be thought of in this way:  There are various possible values of a discrete random variable  The values that have the higher probabilities are the ones that occur more often ●The mean of a probability distribution can be thought of in this way:  There are various possible values of a discrete random variable  The values that have the higher probabilities are the ones that occur more often  The values that occur more often should have a larger role in calculating the mean ●The mean of a probability distribution can be thought of in this way:  There are various possible values of a discrete random variable  The values that have the higher probabilities are the ones that occur more often  The values that occur more often should have a larger role in calculating the mean  The mean is the weighted average of the values, weighted by the probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 23 of 34 Chapter 11 – Section 1 ●The mean of a discrete random variable is μ X = Σ [ x P(x) ] ●The mean of a discrete random variable is μ X = Σ [ x P(x) ] ●In this formula  x are the possible values of X  P(x) is the probability that x occurs  Σ means to add up these terms for all the possible values x

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 24 of 34 Chapter 11 – Section 1 ●Example of a calculation for the mean xP(x)P(x) ●Example of a calculation for the mean xP(x)P(x)x P(x) Multiply ●Example of a calculation for the mean xP(x)P(x)x P(x) Multiply Multiply again ●Example of a calculation for the mean ●Add: = 2.5 ●The mean of this discrete random variable is 2.5 Multiply again

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 25 of 34 Chapter 11 – Section 1 ●The calculation for this problem written out μ X = Σ [ x P(x) ] = [1 0.2] + [2 0.6] + [5 0.1] + [6 0.1] = = 2.5 ●The mean of this discrete random variable is 2.5

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 26 of 34 Chapter 11 – Section 1 ●The mean can also be thought of this way (as in the Law of Large Numbers)  If we repeat the experiment many times ●The mean can also be thought of this way (as in the Law of Large Numbers)  If we repeat the experiment many times  If we record the result each time ●The mean can also be thought of this way (as in the Law of Large Numbers)  If we repeat the experiment many times  If we record the result each time  If we calculate the mean of the results (this is just a mean of a group of numbers) ●The mean can also be thought of this way (as in the Law of Large Numbers)  If we repeat the experiment many times  If we record the result each time  If we calculate the mean of the results (this is just a mean of a group of numbers)  Then this mean of the results gets closer and closer to the mean of the random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 27 of 34 Chapter 11 – Section 1 ●Learning objectives  Distinguish between discrete and continuous random variables  Identify discrete probability distributions  Construct probability histograms  Compute and interpret the mean of a discrete random variable  Interpret the mean of a discrete random variable as an expected value  Compute the variance and standard deviation of a discrete random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 28 of 34 Chapter 11 – Section 1 ●The expected value of a random variable is another term for its mean ●The term “expected value” illustrates the long term nature of the experiments – as we perform more and more experiments, the mean of the results of those experiments gets closer to the “expected value” of the random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 29 of 34 Chapter 11 – Section 1 ●Learning objectives  Distinguish between discrete and continuous random variables  Identify discrete probability distributions  Construct probability histograms  Compute and interpret the mean of a discrete random variable  Interpret the mean of a discrete random variable as an expected value  Compute the variance and standard deviation of a discrete random variable

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 30 of 34 Chapter 11 – Section 1 ●The variance of a discrete random variable is computed similarly as for the mean ●The mean is the weighted sum of the values μ X = Σ [ x P(x) ] ●The variance of a discrete random variable is computed similarly as for the mean ●The mean is the weighted sum of the values μ X = Σ [ x P(x) ] ●The variance is the weighted sum of the squared differences from the mean σ X 2 = Σ [ (x – μ X ) 2 P(x) ] ●The variance of a discrete random variable is computed similarly as for the mean ●The mean is the weighted sum of the values μ X = Σ [ x P(x) ] ●The variance is the weighted sum of the squared differences from the mean σ X 2 = Σ [ (x – μ X ) 2 P(x) ] ●The standard deviation, as we’ve seen before, is the square root of the variance … σ X = √ σ X 2

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 31 of 34 Chapter 11 – Section 1 ●The variance formula σ X 2 = Σ [ (x – μ X ) 2 P(x) ] can involve calculations with many decimals or fractions ●An equivalent formula is σ X 2 = [ Σ x 2 P(x) ] – μ X 2 ●This formula is often easier to compute

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 32 of 34 Chapter 11 – Section 1 ●For variables and samples, we had the concept of a population variance (for the entire population) and a sample variance (for a sample from that population) ●These probability distributions model the complete population  These are population variance formulas  There is no analogy for sample variance here

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 33 of 34 Chapter 11 Section 2 The Binomial Probability Distribution

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 34 of 34 Chapter 11 – Section 2 ●Learning objectives  Determine whether a probability experiment is a binomial experiment  Compute probabilities of binomial experiments  Compute the mean and standard deviation of a binomial random variable  Construct binomial probability histograms

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 35 of 34 Chapter 11 – Section 2 ●Learning objectives  Determine whether a probability experiment is a binomial experiment  Compute probabilities of binomial experiments  Compute the mean and standard deviation of a binomial random variable  Construct binomial probability histograms

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 36 of 34 Chapter 11 – Section 2 ●A binomial experiment has the following structure  The first test is performed … the result is either a success or a failure ●A binomial experiment has the following structure  The first test is performed … the result is either a success or a failure  The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same ●A binomial experiment has the following structure  The first test is performed … the result is either a success or a failure  The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same  A third test is performed … the result is either a success or a failure. The result is independent of the first two and the chance of success is the same

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 37 of 34 Chapter 11 – Section 2 ●Example  A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit ●Example  A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit  The card is then put back into the deck ●Example  A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit  The card is then put back into the deck  A second card is drawn from the deck with the same definition of success. ●Example  A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit  The card is then put back into the deck  A second card is drawn from the deck with the same definition of success.  The second card is put back into the deck ●Example  A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit  The card is then put back into the deck  A second card is drawn from the deck with the same definition of success.  The second card is put back into the deck  We continue for 10 cards

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 38 of 34 Chapter 11 – Section 2 ●A binomial experiment is an experiment with the following characteristics  The experiment is performed a fixed number of times, each time called a trial ●A binomial experiment is an experiment with the following characteristics  The experiment is performed a fixed number of times, each time called a trial  The trials are independent ●A binomial experiment is an experiment with the following characteristics  The experiment is performed a fixed number of times, each time called a trial  The trials are independent  Each trial has two possible outcomes, usually called a success and a failure ●A binomial experiment is an experiment with the following characteristics  The experiment is performed a fixed number of times, each time called a trial  The trials are independent  Each trial has two possible outcomes, usually called a success and a failure  The probability of success is the same for every trial

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 39 of 34 Chapter 11 – Section 2 ●Notation used for binomial distributions  The number of trials is represented by n  The probability of a success is represented by p  The total number of successes in n trials is represented by X ●Because there cannot be a negative number of successes, and because there cannot be more than n successes (out of n attempts) 0 ≤ X ≤ n

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 40 of 34 Chapter 11 – Section 2 ●In our card drawing example  Each trial is the experiment of drawing one card  The experiment is performed 10 times, so n = 10 ●In our card drawing example  Each trial is the experiment of drawing one card  The experiment is performed 10 times, so n = 10  The trials are independent because the drawn card is put back into the deck ●In our card drawing example  Each trial is the experiment of drawing one card  The experiment is performed 10 times, so n = 10  The trials are independent because the drawn card is put back into the deck  Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else  The probability of success is 0.25, the same for every trial, so p = 0.25 ●In our card drawing example  Each trial is the experiment of drawing one card  The experiment is performed 10 times, so n = 10  The trials are independent because the drawn card is put back into the deck  Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else  The probability of success is 0.25, the same for every trial, so p = 0.25  X, the number of successes, is between 0 and 10

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 41 of 34 Chapter 11 – Section 2 ●The word “success” does not mean that this is a good outcome or that we want this to be the outcome ●A “success” in our card drawing experiment is to draw a heart ●If we are counting hearts, then this is the outcome that we are measuring ●There is no good or bad meaning to “success”

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 42 of 34 Chapter 11 – Section 2 ●Learning objectives  Determine whether a probability experiment is a binomial experiment  Compute probabilities of binomial experiments  Compute the mean and standard deviation of a binomial random variable  Construct binomial probability histograms

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 43 of 34 Chapter 11 – Section 2 ●We would like to calculate the probabilities of X, i.e. P(0), P(1), P(2), …, P(n) ●Do a simpler example first  For n = 3 trials  With p =.4 probability of success  Calculate P(2), the probability of 2 successes

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 44 of 34 Chapter 11 – Section 2 ●For 3 trials, the possible ways of getting exactly 2 successes are  S S F  S F S  F S S ●For 3 trials, the possible ways of getting exactly 2 successes are  S S F  S F S  F S S ●The probabilities for each (using the multiplication rule) are  =  =  = 0.096

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 45 of 34 Chapter 11 – Section 2 ●The total probability is P(2) = = ●But there is a pattern  Each way had the same probability … the probability of 2 success (0.4 times 0.4) times the probability of 1 failure (0.6) ●The probability for each case is

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 46 of 34 Chapter 11 – Section 2 ●There are 3 cases  S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second  S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third  F S S could represent choosing a third combination of 2 out of 3 ●There are 3 cases  S S F could represent choosing a combination of 2 out of 3 … choosing the first and the second  S F S could represent choosing a second combination of 2 out of 3 … choosing the first and the third  F S S could represent choosing a third combination of 2 out of 3 ●These are the 3 = 3 C 2 ways to choose 2 out of 3

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 47 of 34 Chapter 11 – Section 2 ●Thus the total probability P(2) = =.288 can also be written as P(2) = 3 C ●Thus the total probability P(2) = =.288 can also be written as P(2) = 3 C ●In other words, the probability is  The number of ways of choosing 2 out of 3, times  The probability of 2 successes, times  The probability of 1 failure

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 48 of 34 Chapter 11 – Section 2 ●The general formula for the binomial probabilities is just this ●For P(x), the probability of x successes, the probability is  The number of ways of choosing x out of n, times  The probability of x successes, times  The probability of n-x failures ●The general formula for the binomial probabilities is just this ●For P(x), the probability of x successes, the probability is  The number of ways of choosing x out of n, times  The probability of x successes, times  The probability of n-x failures ●This formula is P(x) = n C x p x (1 – p) n-x

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 49 of 34 Chapter 11 – Section 2 ●Example ●A student guesses at random on a multiple choice quiz  There are n = 10 questions in total  There are 5 choices per question so that the probability of success p = 1/5 =.2 ●What is the probability that the student gets 6 questions correct?

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 50 of 34 Chapter 11 – Section 2 ●Example continued ●This is a binomial experiment  There are a finite number n = 10 of trials  Each trial has two outcomes (a correct guess and an incorrect guess)  The probability of success is independent from trial to trial (every one is a random guess)  The probability of success p =.2 is the same for each trial

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 51 of 34 Chapter 11 – Section 2 ●Example continued ●The probability of 6 correct guesses is P(x)= n C x p x (1 – p) n-x ●Example continued ●The probability of 6 correct guesses is P(x)= n C x p x (1 – p) n-x = 6 C = = ●Example continued ●The probability of 6 correct guesses is P(x)= n C x p x (1 – p) n-x = 6 C = = ●This is less than a 1% chance ●Example continued ●The probability of 6 correct guesses is P(x)= n C x p x (1 – p) n-x = 6 C = = ●This is less than a 1% chance ●In fact, the chance of getting 6 or more correct (i.e. a passing score) is also less than 1%

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 52 of 34 Chapter 11 – Section 2 ●Binomial calculations can be difficult because of the large numbers (the n C x ) times the small numbers (the p x and (1-p) n-x ) ●It is possible to use tables to look up these probabilities ●It is best to use a calculator routine or a software program to compute these probabilities

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 53 of 34 Chapter 11 – Section 2 ●Learning objectives  Determine whether a probability experiment is a binomial experiment  Compute probabilities of binomial experiments  Compute the mean and standard deviation of a binomial random variable  Construct binomial probability histograms

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 54 of 34 Chapter 11 – Section 2 ●We would like to find the mean of a binomial distribution ●Example  There are 10 questions  The probability of success is.20 on each one  Then the expected number of successes would be = 2 ●The general formula μ X = n p

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 55 of 34 Chapter 11 – Section 2 ●We would like to find the standard deviation and variance of a binomial distribution ●This calculation is more difficult ●The standard deviation is σ X = √ n p (1 – p) and the variance is σ X 2 = n p (1 – p)

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 56 of 34 Chapter 11 – Section 2 ●For our random guessing on a quiz problem  n = 10  p =.2  x = 6 ●For our random guessing on a quiz problem  n = 10  p =.2  x = 6 ●Therefore  The mean is np = 10.2 = 2  The variance is np(1-p) = =.16  The standard deviation is √.16 =.4 ●For our random guessing on a quiz problem  n = 10  p =.2  x = 6 ●Therefore  The mean is np = 10.2 = 2  The variance is np(1-p) = =.16  The standard deviation is √.16 =.4 ● Remember the empirical rule? A passing grade of 6 is 10 standard deviations from the mean …

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 57 of 34 Chapter 11 – Section 2 ●Learning objectives  Determine whether a probability experiment is a binomial experiment  Compute probabilities of binomial experiments  Compute the mean and standard deviation of a binomial random variable  Construct binomial probability histograms

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 58 of 34 Chapter 11 – Section 2 ●With the formula for the binomial probabilities P(x), we can construct histograms for the binomial distribution ●There are three different shapes for these histograms  When p <.5, the histogram is skewed right  When p =.5, the histogram is symmetric  When p >.5, the histogram is skewed left

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 59 of 34 Chapter 11 – Section 2 ●For n = 10 and p =.2 (skewed right)  Mean = 2  Standard deviation =.4

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 60 of 34 Chapter 11 – Section 2 ●For n = 10 and p =.5 (symmetric)  Mean = 5  Standard deviation =.5

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 61 of 34 Chapter 11 – Section 2 ●For n = 10 and p =.8 (skewed left)  Mean = 8  Standard deviation =.4

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 62 of 34 Chapter 11 – Section 2 ●Despite binomial distributions being skewed, the histograms appear more and more bell shaped as n gets larger ●This will be important!

Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 63 of 34 Summary: Chapter 11 – Section 2 ●Binomial random variables model a series of independent trials, each of which can be a success or a failure, each of which has the same probability of success ●The binomial random variable has mean equal to np and variance equal to np(1-p)