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Chapter 6 Modeling Random Events: The Normal and Binomial Models Copyright © 2017, 2014 Pearson Education, Inc. Slide 1 Chapter 6 Topics • Distinguish between discrete and continuous random variables • Apply a Normal model to find probabilities • Apply a binomial model to find probabilities Copyright © 2017, 2014 Pearson Education, Inc. Slide 2 Section 6.1 Rock and Wasp. Shutterstock PROBABILITY DISTRIBUTIONS • What Is a Probability Distribution • Distinguish Between Discrete and Continuous Random Variables • Using Probability Distributions to Compute Probabilities Copyright © 2017, 2014 Pearson Education, Inc. Slide 3 Probability Distribution A table or graph that tells us: 1. All the possible outcomes of a random experiment 2. The probability of each outcome Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 4 Probability Distribution: Table Experiment: Rolling a die x 1 2 3 4 5 6 P(X) 1/6 1/6 1/6 1/6 1/6 1/6 What is the probability of rolling a 5 or a 6? Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 5 Probability Distribution: Graph Estimate the probability that a randomly selected reviewer gave this book a three star rating. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 6 Two Types of Numerical Variables 1. Discrete Numerical values that you can list or count Example: Number of siblings, number of songs on an MP3 player 2. Continuous Cannot be listed or counted because they occur over a range of values Example: Height of basketball players, weight of a sandwich Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 7 Discrete or Continuous Label each of these variables as discrete or continuous: 1. Number of cars owned by a household 2. Number of pets owned by a student 3. Time it takes a worker to commute to a job site 4. Height of a building in downtown San Francisco Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 8 Discrete or Continuous Label each of these variables as discrete or continuous: 1. Number of cars owned by a household - discrete 2. Number of pets owned by a student - discrete 3. Time it takes a worker to commute to a job site continuous 4. Height of a building in downtown San Francisco continuous Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 9 Two Features of a Probability Distribution • All P(X) values must be between 0 and 1. • The sum of all P(X) values must equal 1. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 10 Creating a Probability Distribution Suppose you roll a fair 6-sided die. You win $10 if you roll a 5 or a 6. You lose $5 if you roll a 1. For any other outcome you win or lose nothing. Create a table showing the probability distribution for the amount of money you win playing this game. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 11 Probability Distribution: Example There are three possible outcomes: Win $10, lose $5, or win/lose nothing. X = 10, -5, or 0 The probability of rolling a 5 or 6 (and winning $10) is 2/6. The probability of rolling a 1 (and losing $5) is 1/6. You win/lose nothing by rolling a 2, 3, or 4 (probability = 3/6). Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 12 Probability Distribution: Example X -5 0 10 P(X) 1/6 3/6 2/6 Note: All P(X) values are between 0 and 1, and the sum of all P(X) values = 1. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 13 Probability Distributions for Continuous Random Variables Because you cannot list all outcomes for a continuous random variable, we cannot represent the probability distribution using a table. Graphs, called probability density curves, are used instead. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 14 Probability Density Curve: Example This probability density curve shows the probability of wait times before service at a coffee shop. The shaded area represents the probability of a wait time between 0 and 2 minutes. The total area under the curve = 1. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 15 Finding Probabilities: Uniform Distributions A bus arrives at a certain stop every 12 minutes. The graph shows the probability distribution for wait times before the bus arrives. Use the distribution to find the probability of a wait time between 4 and 10 minutes. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 16 Finding Probabilities: Uniform Distributions The shaded area represents the probability of a wait time between 4 and 10 minutes. We can find the area of this rectangle: (length x height). 6 x 0.08333 = 0.49998 ≈ 0.5000 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 17 Finding Areas Under Curves In general, finding areas under curves that are not geometric shapes like rectangles is difficult and requires using some type of computerbased technology. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 18 Irina Mos. Shutterstock Section 6.2 THE NORMAL MODEL • Finding Probabilities Using the Normal Model • Finding Values for a Random Variable Given a Normal Probability Copyright © 2017, 2014 Pearson Education, Inc. Slide 19 The Normal Model • One of the most widely used probability models for continuous numerical random variables • Also called the “bell curve” because of its shape Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 20 Visualizing the Normal Model This histogram shows heights for a sample of males. Since it is symmetric and unimodal, it can be modeled using a Normal distribution. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 21 The Normal Distribution: Same Mean, Different Standard Deviation These two Normal distributions have the same mean but different standard deviations. Note: The area under both curves = 1. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 22 The Normal Distribution: Different Means, Same Standard Deviation These normal curves show the distribution of heights for adult men and women in the US. Notice that the spread of each curve (the standard deviation) is about the same, but that the center of each curve (the mean) is different. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 23 Notation • We use the symbol μ (“mu”) to represent the mean of a probability distribution. • We use the symbol σ (“sigma”) to represent the standard deviation of a probability distribution. • The notation N(μ,σ) represents a Normal distribution with mean μ and standard deviation σ. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 24 Finding Probability with Technology • It is always helpful to sketch a picture of the area you are interested in finding before using technology to find the area. • Most technologies require you to enter the mean and standard deviation of the distribution, as well as the beginning and ending x-values of the area you are finding in the Normal model. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 25 Using the TI-84 Calculator To find a probability for a Normal distribution on the TI-84 calculator: 1. Push 2ND Dist then select option 2: normalcdf. 2. Enter the left boundary of your shaded area, the right boundary of your shaded area, the mean, and the standard deviation and press ENTER. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 26 Example: Women’s Heights The Normal model N(64, 3) gives a good approximation of adult women’s heights (in inches) in the US. Find the probability that a randomly selected adult woman from the US is between 62 and 67 inches tall. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 27 Example: Women’s Heights To find this area using a TI-84 calculator: normalcdf(62, 67, 64, 3) = 0.589 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 28 Using StatCrunch Research has shown that the distribution of newborn Pacific harbor seals’ birth lengths is approximately Normal with a mean of 29.5 inches and a standard deviation of 2.5 inches. Find the probability that a randomly selected harbor seal pup is longer than 31 inches. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 29 Using StatCrunch 1. 2. 3. 4. Select STAT > Calculators > Normal. Enter the mean and standard deviation. Enter x ≥ 31. Press COMPUTE. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 30 Seal Pups P(x ≥ 31) = 0.274 Copyright © 2017, 2014 Pearson Education, Inc. Slide 31 Without Technology • To find Normal probabilities without technology, we convert the information in the problem to standard units and find the probability using a standard Normal table. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 32 Standard Normal Model The Standard Normal Model is a normal distribution with a mean = 0 and a standard deviation = 1. N(0, 1) Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 33 Standard Units or “Z-Scores” • Tell us how many standard deviations from the mean an observation lies. xx z s Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 34 Seal Pups: Using a Table Small newborn seal pups have a lower chance of survival than larger newborn pups. Suppose the length of newborn seal pups follow a Normal distribution with a mean of 29.5 inches and a standard deviation of 1.2 inches. What is the probability that a randomly selected newborn seal pup is shorter than 28.0 inches? Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 35 Seal Pups: Using a Table Convert the information in the problem into a standard score or z-score: 28 - 29.5 -1.5 z= = = -1.25 1.2 1.2 Now look up –1.25 on the Normal table. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 36 Going “Backwards”: Finding x-values Given a Probability Sometimes we are interested in finding the value for the random variable associated with a given probability. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 37 Example: Seal Pups The birth lengths of newborn seal pups follow the model N(29.5, 1.2). Suppose pups with lengths at the 15th percentile and below are unlikely to survive. What birth lengths would make it unlikely that a pup would survive? Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 38 Using StatCrunch The shaded area = 0.15. The x-value that corresponds with the 15th percentile is x = 28.3 inches. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 39 Using the TI-84 Calculator To find an x-value for a Normal distribution on the TI-84 calculator given an area: 1. Push 2nd DIST then select option 3: invNorm. 2. Enter the area to the left of the x-value of interest, the mean, and the standard deviation. To find the 15th percentile for the seal pup lengths, invNorm(0.15, 29.5, 1.2) = 28.3 inches. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 40 Normal or Inverse Normal? It is known that the heights of adult men follow a Normal model. For each of the following situations, decide if it is a Normal or Inverse Normal problem. 1. A clothing store manager wonders what percentage of her customers are taller than six feet. 2. A clothing store manager wants to cater to the tallest 20% of men and wants to know what heights she should accommodate. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 41 Normal or Inverse Normal? 1. A clothing store manager wonders what percentage of her customers are taller than six feet. This is a Normal problem. 2. A clothing store manager wants to cater to the tallest 20% of men and wants to know what heights she should accommodate. This is an Inverse Normal problem. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 42 Svetlana Larina. Shutterstock Section 6.3 THE BINOMIAL MODEL • Finding Probabilities Using the Binomial Model Copyright © 2017, 2014 Pearson Education, Inc. Slide 43 Binomial Probability Model • Useful for many situations with discrete numerical variables Look for: 1. A fixed number of trials (n). 2. Only two outcomes possible at each trial. 3. The probability (p) of a success is the same at each trial. 4. The trials are independent. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 44 Example: Tossing a Coin Suppose you toss a coin eight times and count the number of heads. Explain why this is an example of a binomial experiment. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 45 Binomial Experiment: Tossing a Coin 1. There is a fixed number of trials (8). 2. Each trial has only two outcomes (heads or tails). 3. The probability of a landing on heads is the same at each trial (1/2). 4. Trials are independent (outcome of one toss does not affect the outcome of any other toss). Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 46 Binomial or Not Binomial For each of the following situations, identify which are binomial. For the ones that are not binomial, explain why it is not binomial. 1. Record the ages of a group of 20 randomly selected statistics students. 2. Ask each partner of a married couple whether or not they exercise regularly. 3. Ask a random sample of workers whether their annual salary is greater than $50,000. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 47 Binomial or Not Binomial 1. Record the ages of a group of 20 randomly selected statistics students. Not binomial (each trial does not have only two possible outcomes). 2. Ask each partner of a married couple whether or not they exercise regularly. Not binomial (since the couple is married, the trials are not independent). 3. Ask a random sample of workers whether their annual salary is greater than $50,000. Binomial Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 48 Shape of a Binomial Distribution • Depends on both n and p Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 49 Shape of a Binomial Distribution • Symmetric if p = 0.5, but also when n is large even if p is close to 0 or 1. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 50 Finding Binomial Probabilities • Binomial probabilities can be found by hand using the Binomial formula if n and x are relatively small. • In most applications, technology that has the binomial distribution built in is used to find binomial probabilities. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 51 Example: Finding Binomial Probabilities for 1 Value of x According to a recent Pew poll, 47% of American say it is important to control gun ownership. In a random sample of 200 Americans, find the probability that exactly 90 Americans say it is important to control gun ownership. Source: www.Pewresearch.org Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 52 Example: Gun Control Check that this is a binomial problem: 1. Fixed number of trials (n = 200) 2. Only two outcomes for each trial (Is it important to control gun ownership? Yes or no) 3. Probability of a “success” remains the same for each trial (p = 0.47) 4. Trials are independent Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 53 Notation: Binomial Probability In this problem we are looking for the binomial probability where n = 200, p = 0.47, and x = 90. We use the notation b(200, 0.47, 90) to denote this probability. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 54 Finding Binomial Probabilities: Single Value of x Using StatCrunch Use STAT > Calculators > Binomial Enter n, p, and x=90. P(X=90) = 0.048 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 55 Using the TI-84 Calculator To find a binomial probability for a single value of x on the TI-84 calculator: 1. Push 2ND Dist then select option binompdf. 2. Enter binompdf(n, p, x). binompdf(200, 0.47, 90) = 0.048 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 56 Finding Binomial Probabilities for More Than One Value of x According to the Bureau of Labor Statistics, the unemployment rate in Detroit in June 2014 was 10.2%. If a random sample of 15 people living in Detroit at that time was taken, find the probability that three or fewer were unemployed. Source: www.bls.gov Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 57 Example: Unemployment in Detroit This problem is binomial because: 1. There is a fixed number of trials (n = 15). 2. There are only two outcomes for each trial (Unemployed? Yes/no). 3. The probability of a “success” is the same for each trial (p = 0.102). 4. The trials are independent. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 58 Example: Unemployment in Detroit n = 15, p = 0.102 “Three or fewer” means x = 0, 1, 2, or 3 We need to compute: b(15, 0.102, 0)+ b(15, 0.102,1)+ b(15, 0.102, 2)+ b(15, 0.102,3) Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 59 Probability Cumulative Density Functions Instead of computing four binomial probabilities separately and adding them together, we can take advantage of the probability cumulative density functions built into certain technologies. These functions add together probabilities for a range of x values. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 60 Binomial Cumulative Density Function: StatCrunch In the unemployment problem for a random sample of 15 people from Detroit in June 2014, n = 15, p = 0.102, and x = 0, 1, 2, 3. In StatCrunch, use STAT > Calculators > Binomial Enter n, p, and set x ≤ 3. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 61 Cumulative Probabilities Using StatCrunch The sum of the red bars represents P(X≤3). P(X≤3) = 0.941 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 62 Using the TI-84 Calculator To find a binomial probability involving more than 1 value of x on the TI-84 calculator: 1. Push 2ND Dist then select option binomcdf. 2. Enter binomcdf(n, p, x). NOTE: The binomcdf command always adds probabilities starting with x = 0. Finding the sum of probabilities for x values not starting at 0 requires subtracting several binomcdf calculations. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 63 Binomial Cumulative Density Function: TI-84 Calculator In the unemployment problem for a random sample of 15 people from Detroit in June 2014, n = 15, p = 0.102, and x = 0, 1, 2, 3. NOTE: We want to add probabilities starting with x = 0, so we can use the binomcdf command. binomcdf(15, 0.102, 3) = 0.941 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 64 Being Careful with Language Because the binomial probability distribution models the probability of discrete random variables, we have to pay close attention to language. “More than 5” mean x = 6, 7, 8, …, n “5 or more” mean x = 5, 6, 7, 8, …, n NOTE: The second case includes b(n, p, 5) in the sum while the first case does not. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 65 A Difference between the Normal and Binomial Distributions We did not need to worry about this distinction with Normal probabilities because for continuous numerical variables, the probability of getting five or more is the same as the probability of getting more than five. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 66 Practice Identifying X-Values What x value(s) are implied in each of these phrases? 1. More than 6 2. Fewer than 5 3. At least 7 4. Between 3 and 7 (inclusive) 5. 10 or more Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 67 Practice Identifying X-Values 1. 2. 3. 4. 5. More than 6 x = 7, 8, 9, …, n Fewer than 5 x = 0, 1, 2, 3, 4 At least 7 x = 7, 8, 9, …, n Between 3 and 7 (inclusive) x = 3, 4, 5, 6, 7 10 or more x = 10, 11, 12, …, n Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 68 Example: Medical Coverage According to a report in the Sacramento Bee, only 11% of Californians lacked medical insurance after the passage of the Affordable Care Act (down from a previous rate of 17%). Suppose a sample of 80 Californians was taken after the passage of the Affordable Care Act. Find the probability that: a. More than 15 lack medical insurance. b. Between 10 and 20 (inclusive) lack medical insurance. Source: www.sacbee.com Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 69 Example: Medical Coverage a. More than 15 lack medical insurance n = 80, p = 0.11, x =16, 17, …, 80 NOTE: x > 15 P(X>15) = 0.013 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 70 Example: Medical Coverage b. Between 10 and 20 (inclusive) lack medical insurance. n = 80, p = 0.11, x = 10, 11, …, 20 P(10≤x≤20) = 0.384 Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 71 The Mean of a Binomial Distribution • Sometimes called the “expected value” • Can be thought of as the number of “successes” we would expect if the experiment were repeated n times • As with all distributions, tells us where the distribution “balances” • Formula: m = np Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 72 Example: Mean of a Binomial Experiment Suppose you toss a fair coin 15 times and count the number of heads. Find and interpret the mean of the probability distribution. Mean = 15(0.5) = 7.5 We expect between 7 and 8 heads in 15 tosses. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 73 Example: Mean of a Binomial Experiment Note: The mean represents the balancing point of the distribution. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 74 The Standard Deviation of a Binomial Distribution • Measures the spread of a distribution • Formula: s = np(1- p) • Can help us approximate a range of values we would expect and a range of values we would find surprising in a binomial distribution Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 75 Example: Standard Deviation of a Binomial Distribution During his career, major league baseball player Buster Posey of the San Francisco Giants has a batting average of 0.308. This means that he gets a hit about 30.8% of the time when he bats. If he bats 480 times in a season, we would expect him to get, give or take, how many hits? Would it be unusual if he got 120 hits? Source: www.baseball-reference.com Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 76 Example: Buster Posey Expected value: μ = 480(0.308) = 147.8 Standard deviation: 480(0.308)(1 0.308) 10.1 We expect Posey to get 147.8 ± 10.1 hits. 147.8 – 10.1 = 137.7 147.8 + 10.1 = 157.9 We expect Posey to get between 137.7 – 157.9 hits. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 77 Example: Buster Posey Would it be unusual if he got 120 hits? We know 120 is beyond one standard deviation below the mean. To determine if this would be surprising we can compute a z-score: 120 -147.8 z= = -2.75 10.1 Yes, this would be surprising since is it over two standard deviations below the mean. Copyright Copyright©©2017, 2017,2014 2014Pearson PearsonEducation, Education,Inc. Inc. Slide 78