Distributions of Continuous Data

Transcription

1 C H A P T ER Distributions of Continuous Data New cars and trucks sold in the United States average about 28 highway miles per gallon (mpg) in 2010, up from about 24 mpg in Some of the improvement is due to the proliferation of hybrid cars, which can achieve over 50 mpg on the highway. You will analyze the fuel efficiency of two types of hybrid cars..1 Products and Probabilities Discrete Data and Probability Distributions p Catching Some Z s? Z-Scores and the Standard Normal Distribution p Basketball and Blood Type The Binomial Probability Distribution p Above and In-Between Probabilities Above and Between Z-Scores p Charge It! Continuous Data and the Normal Probability Distribution p The Old, The News, and Making the Grade Applications of the Normal Distribution p Recharge It! The Standard Normal Probability Distribution p. 793 Chapter l Distributions of Continuous Data 753

2 754 Chapter l Distributions of Continuous Data

3 .1 Products and Probabilities Discrete Data and Probability Distributions Objectives In this lesson you will: l Display probability distributions for discrete data as relative frequency tables and probability histograms. l Interpret probability distributions. l Calculate theoretical probabilities. l Compare theoretical and experimental probabilities. Key Terms l discrete data l continuous data l probability distribution l relative frequency table l probability histogram l experimental probability l theoretical probability Problem 1 The Product Game Concepts of probability are used to design board games, hand-held games, and video games. Gaming has become so popular that many colleges now offer majors in Computer and Video Game Design, Computer Game Programming, and Computer Game Production. Let s look at a few games. The Product Game has the following rules. l For each round, Player 1 rolls 2 number cubes while Player 2 records the product of the 2 numbers displayed on the number cubes. l Player 1 wins the round if the product ends in an even number. l Player 2 wins the round if the product ends in an odd number. l The player with the most winning rounds is the winner of the Product Game. 1. Which player do you think will win the Product Game? Why? Lesson.1 l Discrete Data and Probability Distributions 755

4 2. With a partner, decide who will be Player 1 and who will be Player 2. Play 20 rounds of the Product Game. For each round, record the product and the winner of the round. Round Number Product Winner 3. Who won the Product Game? Discrete data is data that has a finite number of possible values. The products that you recorded in the Product Game are examples of discrete data. The product will always be an integer from 1 to 36. Continuous data is data that has an infinite number of possible values. Heights of students, time to complete a test, and distance between two cities are examples of continuous data. You will work more with continuous data later in this chapter. A set of discrete data values can be plotted as a set of single points on a number line. A set of continuous data values can be plotted as an interval on a number line. Example: discrete data continuous data 756 Chapter l Distributions of Continuous Data

5 4. Create a stem and leaf plot to summarize the products from Question 2. Let the stem represent the last digit of each product and the leaf represent the tens digit of each product. So, the key is How does the stem and leaf plot make it easier to determine who won the Product Game? 6. Is a stem and leaf plot appropriate to display the products for the entire class? Why or why not? 7. What type of graph could be used to display the products for the entire class? Explain. Lesson.1 l Discrete Data and Probability Distributions 757

6 8. Create a frequency table to summarize the products for the entire class. Last Digit Number of Occurrences 9. Create a histogram to summarize the products for the entire class. A probability distribution provides all possible discrete values that can occur and the probability of each. Probability distributions for discrete data can be displayed as a table, a graph, or an equation. A relative frequency table provides all possible discrete values that can occur in one column and the probability of each possible value in another column. A probability histogram displays possible discrete values on the horizontal axis and the probability of each possible value on the vertical axis. 758 Chapter l Distributions of Continuous Data

7 10. Create a relative frequency table to summarize the products for the entire class. Last Digit 0 Number of Occurrences Relative Frequency Create a probability histogram to summarize the products for the entire class. Lesson.1 l Discrete Data and Probability Distributions 759

8 12. What do you notice about the shapes of the frequency histogram in Question 9 and the probability histogram in Question 11? 13. Calculate each probability using the relative frequency table from Question 10. a. Probability of rolling a product with an even last digit. b. Probability of rolling a product with an odd last digit. 14. Based on your calculations, which player has a better chance of winning? Explain. Problem 2 Theoretical Probability Experimental probability is the chance that something happens based on repeating experiments and observing the outcomes. The probabilities that you calculated in Problem 1 are experimental probabilities because they were based on an experiment of rolling two number cubes and calculating the product. Theoretical probability is a mathematical calculation that an event will happen in theory. 1. Complete the table to summarize the possible products of the numbers displayed on two number cubes. No. Cube No. Cube Chapter l Distributions of Continuous Data

9 2. Create a frequency table to summarize the last digit of the possible products. Last Digit Number of Occurrences 3. Create a relative frequency table to summarize the last digit of the possible products. Last Digit 0 Number of Occurrences Relative Frequency Lesson.1 l Discrete Data and Probability Distributions 761

10 4. Create a probability histogram to summarize the last digit of the possible products. 5. Calculate each probability using the relative frequency table from Question 3. a. Theoretical probability of rolling a product with an even last digit. b. Theoretical probability of rolling a product with an odd last digit. 6. Based on the theoretical probabilities, which player has a better chance of winning? Explain. 7. Is the Product Game a fair game? Explain. 762 Chapter l Distributions of Continuous Data

11 8. Compare the theoretical probabilities to the experimental probabilities for the Product Game. Why are they not exactly the same? 9. Use the theoretical probabilities to make each prediction. a. If you roll two number cubes 100 times, how many times will the last digit of the product be a 5? b. If you roll two number cubes 500 times, how many times will the last digit of the product be a 2? c. If you roll two number cubes 10,000 times, how many times will the last digit of the product be a 7? Problem 3 Penny Toss The Penny Toss Game has the following rules. For each round, Player 1 tosses a penny 3 times while Player 2 records the total number of heads. l Player 1 wins if the penny shows heads 0 or 1 times. l Player 2 wins if the penny shows heads 2 or 3 times. l The player with the most winning rounds is the winner of the Penny Toss Game. 1. Which player do you think will win the Penny Toss Game? Explain. Lesson.1 l Discrete Data and Probability Distributions 763

12 2. With a partner, decide who will be Player 1 and who will be Player 2. Play 20 rounds of the Penny Toss Game. For each round, record the number of heads and the winner of the round. Trial No. Number of Heads Player 1 Player Create a relative frequency table to summarize the number of heads in each round. Outcome 0 Heads 1 Head 2 Heads 3 Heads Frequency Probability 764 Chapter l Distributions of Continuous Data

13 4. Create a probability histogram to summarize the number of heads in each round. 5. Create a tree diagram to summarize the possible number of heads when flipping a penny three times. Lesson.1 l Discrete Data and Probability Distributions 765

14 6. Create a relative frequency table to summarize the possible number of heads when flipping a penny three times. Outcome 0 Heads 1 Head 2 Heads 3 Heads Frequency Probability 7. Create a probability histogram to summarize the possible number of heads when flipping a penny three times. 8. Compare the experimental and theoretical probability distributions and histograms. Why are they not exactly the same? 9. Is the Penny Toss Game a fair game? Explain. Be prepared to share your methods and solutions. 766 Chapter l Distributions of Continuous Data

15 .2 Basketball and Blood Type The Binomial Probability Distribution Objectives In this lesson you will: l Identify and apply properties of binomial probability distributions. l Identify conditions necessary for an experiment to be considered binomial. l Calculate binomial probabilities. Key Terms l binomial experiment l binomial probability distribution l cumulative probability Problem 1 Basketball Bash Brianna is the leading free thrower on the girls basketball team, making 80% of her free throws. At the interscholastic Basketball Bash, each high school sends one player to the foul line to shoot free throws. Trophies are awarded for the most consecutive free throws and the highest percentage of free throws made. 1. Do you think Brianna has a better chance of making 5 free throws in a row or at least 7 out of 10 free throws? Explain. An experiment is a binomial experiment if the following conditions are met. l There are a fixed number of trials or repetitions of the experiment. Let n represent the number of trials. l Each trial is independent of every other trial. l Each trial has two mutually exclusive outcomes. Success has a probability of p. Failure has a probability of 1 p. l The probability of success is the same for each trial. Lesson.2 l The Binomial Probability Distribution 767

16 2. Is the Basketball Bash a binomial experiment? Explain. 3. Determine whether each experiment is a binomial experiment. If the experiment is binomial, identify the number of trials n, the possible outcomes, the probability of success p, and the probability of failure 1 p. If the experiment is not binomial, explain why not. a. Two number cubes are rolled 10 times and the number of times the product of the dots shown is a prime number is recorded. b. Two number cubes are rolled 10 times and the sum of the dots is recorded. 768 Chapter l Distributions of Continuous Data

17 c. A random sample of 10 people in the U.S. is taken and the number who have each blood type is recorded. d. A random sample of 5 people in the U.S. is taken and the number of people with type O-negative blood is recorded. (According to the American Red Cross, only 7% of the U.S. population has type O-negative blood.) The binomial probability distribution describes probabilities for outcomes of multiple trials of binomial experiments. To identify the possible outcomes over multiple trials, a table or tree diagram is often useful. 4. Determine the probability distribution for 2 free throw attempts by Brianna. a. Create a tree diagram to summarize the possible outcomes for two free throw attempts. b. List the possible outcomes for two free throw attempts. Lesson.2 l The Binomial Probability Distribution 769

18 c. Calculate the probability of each outcome. Outcome Probability of Outcome in Exponential Form Probability of Outcome SS d. Complete the table to define the probability distribution for two free throw attempts by Brianna. Number of Free Throws Made Probability 5. Calculate each probability. a. Probability that Brianna makes both free throws, P(makes two free throws). b. Probability that Brianna makes exactly one of two free throws, P(makes exactly one free throw). c. Probability that Brianna makes at least one of the two free throws, P(makes at least one free throw). 6. Determine the probability distribution for 3 free throw attempts by Brianna. a. Create a tree diagram to summarize the possible outcomes for three free throw attempts. 770 Chapter l Distributions of Continuous Data

19 b. List the possible outcomes for three free throw attempts. c. Calculate the probability of each outcome. Outcome Probability of Outcome in Exponential Form Probability of Outcome SSS d. Complete the table to define the probability distribution for three free throw attempts by Brianna. Number of Free Throws Made Probability Calculate each probability. a. Probability that Brianna makes all three free throws, P(makes three free throws). b. Probability that Brianna makes exactly two of three free throws, P(makes exactly two free throws). c. Probability that Brianna makes at least two of the three free throws, P(makes at least two free throws). d. Probability that Brianna makes exactly one of three free throws, P(makes exactly one free throw). Lesson.2 l The Binomial Probability Distribution 771

20 e. Probability that Brianna makes at least one of the three free throws, P(makes at least one free throw). 8. What patterns do you notice in the probability distribution? 9. Compare the binomial probability distributions for 2 free throws and 3 free throws. What patterns do you notice? Take Note The combination of n objects taken x at a time is calculated as x!(n x)!. n C x n! For a binomial experiment with n trials, the probability of x successful trials is P(x) n C x p x (1 p) n x n! x!(n x)! px (1 p) n x where x is the number of successful trials, n is the number of trials, p is the probability of success, and n C x represents the number of ways of obtaining x successes in n trials. 10. Calculate each probability for 3 free throws by Brianna using the formula. Verify that the probabilities are the same as what you calculated in Question 6, part (d). a. P(0) 772 Chapter l Distributions of Continuous Data

21 b. P(1) c. P(2) d. P(3) 11. Calculate each probability. a. The probability of Brianna making exactly 5 of 8 free throws. b. The probability of Brianna making exactly 19 of 20 free throws. c. The probability of Brianna making exactly 1 of 5 free throws. The binomial probability distribution lists the probabilities for each outcome. A cumulative probability is a probability that collects or adds several probabilities and is often expressed using words such as at least, at most, less than, or greater than. Lesson.2 l The Binomial Probability Distribution 773

22 12. Look at the probabilities that you calculated in Question 5 and Question 7. Which of these probabilities are cumulative probabilities? 13. Calculate each probability if Brianna is attempting 3 free throws. a. P(x 1): b. P(x 1): 14. What is the difference between P(x 1) and P(x 1)? 15. Calculate the probability of Brianna making 2 or fewer free throws in three attempts. 774 Chapter l Distributions of Continuous Data

23 Problem 2 Blood Type The American Red Cross reports that only 7% of the U.S. population has type O-negative blood. During a blood drive sponsored by the Student Government Association, 125 people donate blood. 1. Is the blood drive a binomial experiment? Explain. 2. You want to calculate the probability that 5 people who donated had O-negative blood type. Identify the following. a. The number of trials, n. b. The number of successful trials, x. c. The probability of success, p. 3. What is the probability that 5 people who donated had O-negative blood type? 4. How would you calculate the probability that 5 or fewer people who donated had type O-negative blood? Lesson.2 l The Binomial Probability Distribution 775

24 5. How would you calculate the probability that at least 20 people who donated had type O-negative blood? To calculate a binomial probability using a graphing calculator, perform the following steps. l Press the 2ND button and the VARS button to select the DISTR menu. l Select 0: binompdf(. l Enter values for n, p, and x separated by commas and followed by a right parenthesis. l Press ENTER. 6. Calculate each probability using a graphing calculator. a. The probability that exactly 5 people who donated had type O-negative blood. b. The probability that exactly 10 people who donated had type O-negative blood. c. The probability that exactly 15 people who donated had type O-negative blood. 7. How does your answer to Question 6, part (a) compare to your answer to Question 3? 776 Chapter l Distributions of Continuous Data

25 To calculate the cumulative probability of x or fewer successes using a graphing calculator, perform the following steps. l Press the 2ND button and the VARS button to select the DISTR menu. l Select A: binomcdf(. l Enter values for n, p, and x separated by commas and followed by a right parenthesis. l Press ENTER. 8. How can you calculate the cumulative probability of greater than x successes? 9. Calculate each probability using a graphing calculator. a. The probability 10 of the 125 blood donors had type O-negative blood: b. The probability fewer than 10 of the 125 blood donors had type O-negative blood: c. The probability 10 or fewer of the 125 blood donors had type O-negative blood: d. The probability at least 10 of the 125 blood donors had type O-negative blood: Lesson.2 l The Binomial Probability Distribution 777

26 Problem 3 Basketball Bash Revisited 1. Recall that Brianna s free throw percentage is 80%. Does Brianna have a better chance of making 5 free throws in a row or at least 7 out of 10 free throws? Explain. Be prepared to share your methods and solutions. 778 Chapter l Distributions of Continuous Data

27 .3 Charge It! Continuous Data and the Normal Probability Distribution Objectives In this lesson you will: l Differentiate between discrete and continuous data. l Draw distributions for continuous data. l Construct and interpret properties of the normal curve and the normal probability distribution. l Apply the Empirical Rule for Normal Distributions. Key Terms l normal curve l normal distribution/normal probability distribution l Empirical Rule for Normal Distributions Problem 1 Charge It! One important feature of a cell phone is how long it can be used before requiring charging. Two companies claim their phones, E-Phone and Teaberry, can be used for an average of 7 hours before requiring charging. 1. What does it mean for the phones to have an average time of 7 hours before needing to be charged? 2. Does 7 hours represent the mean, median, or mode? Explain. 3. Is the time a phone can be used before requiring charging discrete data or continuous data? Explain. Lesson.3 l Continuous Data and the Normal Probability Distribution 779

28 Remember continuous data is data that has an infinite number of possible values. It is not possible to calculate the probability of a particular data value because continuous data can take on infinitely many values. Instead, you can calculate the probability of an interval of values. Continuous data is often displayed using a relative frequency histogram with an interval of values, rather than individual values, on the horizontal axis. 4. Each table shows the time before recharging for a random sample of 100 cell phones. Complete the tables shown by calculating the relative frequency for each interval. Time Before Recharging (hours) E-Phone Number of Phones Relative Frequency Time Before Recharging (hours) Teaberry Number of Phones Relative Frequency 780 Chapter l Distributions of Continuous Data

29 5. Create a relative frequency histogram for each phone model. Lesson.3 l Continuous Data and the Normal Probability Distribution 781

30 6. Describe the shape and spread of the histograms. 7. If the random sample included 1000 phones instead of 100, how would the histograms change? If the sample size of phones increased and the interval size decreased, the relative frequency histograms would continue to be bell-shaped and symmetric about the mean. A normal curve is a curve that is bell-shaped and symmetric about the mean. A normal distribution, or normal probability distribution, describes a continuous data set that can be modeled using a normal curve. The normal curve for the following histogram is shown. Many continuous real-world data sets follow a normal distribution, including adult IQ scores, height and weight of males and females at different age levels, gas mileage of certain cars, and SAT and ACT scores. 782 Chapter l Distributions of Continuous Data

31 Data that follows a normal distribution has the following properties. l The relative frequency histogram of the data is bell-shaped. l The mean of the data set is equal to the median of the data set. l The data is symmetric about the mean and median. Half of the data values are above the mean and median. Half of the data values are below the mean and median. l The curvature of the normal curve changes at 1 standard deviation above and below the mean. l The area under the normal curve is equal to 1. 1 standard deviation below the mean Mean 1 standard deviation above the mean Lesson.3 l Continuous Data and the Normal Probability Distribution 783

32 8. The E-Phone has a standard deviation of 0.5 hours. The Teaberry has a standard deviation of 0.4 hours. Label each number line so that the curve is a normal curve and follows the properties of the normal distribution. Include 3 standard deviations above and below the mean. E-Phone Teaberry 9. What are the similarities and differences between the number lines? 784 Chapter l Distributions of Continuous Data

33 Problem 2 The Empire Rules! The area under the normal curve over an interval is equal to the probability of a randomly chosen value falling in that interval. Determining an equation for the normal curve and calculating the area under the curve algebraically are beyond the scope of this course. Luckily, a rule can be used to calculate probabilities over certain intervals. The Empirical Rule for Normal Distributions states that: l Approximately 68% of the area under the normal curve is within one standard deviation of the mean. l Approximately 95% of the area under the normal curve is within two standard deviations of the mean. l Approximately 99.7% of the area under the normal curve is within three standard deviations of the mean. 68% 1 standard deviation below the mean Mean 1 standard deviation above the mean 2 standard deviations below the mean 95% Mean 2 standard deviations above the mean 99.7% 3 standard deviations below the mean Mean 3 standard deviations above the mean Lesson.3 l Continuous Data and the Normal Probability Distribution 785

34 1. E-Phone has a mean time before recharging of 7 hours and a standard deviation of 0.5 hours. Teaberry has a mean time before recharging of 7 hours and a standard deviation of 0.4 hours. The time before recharging for both cell phones follows a normal distribution. a. Approximately 68% of E-Phones will require charging between what 2 times? b. Approximately 68% of Teaberries will require charging between what 2 times? c. Approximately 95% of E-Phones will require charging between what 2 times? d. Approximately 95% of Teaberries will require charging between what 2 times? 2. Which phone would you choose if you were only considering the time before recharging? Explain. 786 Chapter l Distributions of Continuous Data

35 3. The percentages included in the Empirical Rule for Normal Distributions can be used to calculate probabilities for other intervals. For each interval, shade the portion of the normal curve. Then calculate the probability. a. Probability of a data value being above the mean. b. Probability of a data value being between the mean and one standard deviation above the mean. c. Probability of a data value being between the mean and two standard deviations below the mean. Lesson.3 l Continuous Data and the Normal Probability Distribution 787

36 d. Probability of a data value being between one and two standard deviations above the mean. e. Probability of a data value being between two standard deviations below the mean and one standard deviation above the mean. 788 Chapter l Distributions of Continuous Data

37 f. Probability of a data value being above one standard deviation above the mean. g. Probability of a data value being below two standard deviations below the mean. Lesson.3 l Continuous Data and the Normal Probability Distribution 789

38 Problem 3 Summary A school district administered the same math tests to all students in the district. On the midterm test, the average score was 75 and the standard deviation was 5. On the final test, the average score was 80 and the standard deviation was 2. The scores of both tests followed a normal distribution. 1. Label each number line so that the curve is a normal curve and follows the properties of the normal distribution. Include 3 standard deviations above and below the mean. 2. Half of the students in the district scored below what number on the midterm test? On the final test? 3. About 95% of the students in the district scored between what two numbers on the midterm test? On the final test? 790 Chapter l Distributions of Continuous Data

39 4. About 16% of the students in the district scored below what number on the midterm test? On the final test? 5. About 2.5% of the students scored above what number on the midterm test? On the final test? 6. Eduard and Dana both scored 82 on both tests. Eduard said they actually did better on the midterm test. Dana says they did the same on both tests. Who is correct? Be prepared to share your methods and solutions. Lesson.3 l Continuous Data and the Normal Probability Distribution 791

40 792 Chapter l Distributions of Continuous Data

41 .4 Recharge It! The Standard Normal Probability Distribution Objectives In this lesson you will: l Describe the effect of changing the mean and standard deviation on the normal curve. l Apply the properties of the standard normal probability distribution. Key Term l standard normal distribution Problem 1 Recharge It Clay and Justin are goalies on different hockey teams. During the season, each stopped an average of 30 shots per game. Clay has a standard deviation of 5 shots. Justin has a standard deviation of 4 shots. The normal curves for each goalie are shown. Clay Justin Lesson.4 l The Standard Normal Probability Distribution 793

42 1. How are the graphs the same and how are they different? Explain. 2. Normal curves A, B, and C are shown. A B C a. Label the mean of each curve on the number line. b. Which curve has the largest standard deviation? Explain. c. Which curve has the smallest standard deviation? Explain. 794 Chapter l Distributions of Continuous Data

43 3. Normal curves A, B, and C are shown. A B C a. Locate the means of the 3 curves on the number line. b. Which curve has the largest standard deviation? Explain. c. Which curve has the smallest standard deviation? Explain 4. If two normal curves have the same mean but different standard deviations, what will be true about the graphs? 5. If two normal curves have the same standard deviation but different means, what will be true about the graphs? Lesson.4 l The Standard Normal Probability Distribution 795

44 6. Match the following normal distributions with each normal curve. mean 100 mean 100 mean 150 mean 150 standard deviation 10 standard deviation 25 standard deviation 50 standard deviation 25 a b c d Chapter l Distributions of Continuous Data

45 Problem 2 The Standard Normal Distribution As seen in Problem 1, every normal distribution curve can be described as one or more transformations. 1. What type of transformation occurs when the mean changes? 2. What type of transformation occurs when the standard deviation changes? Transformations of other functions can be described in terms of transformations to a parent function, the most basic function for each function family. For example, the parent quadratic function is f(x) x 2. The parent square root function is f(x) x. Does a normal distribution curve also have a parent? The standard normal distribution is a normal probability distribution with the following properties: l The mean is equal to 0. l The standard deviation is 1. l The curve is bell-shaped and symmetric about the mean. All normal distributions you have seen included the units of the data values along the horizontal axis. Examples of units include hours to measure the time it takes for a phone to require recharging or shots to describe a hockey goalie s shots saved per game. The standard normal distribution uses a unit of standard deviations. For example, a particular value may be 1 standard deviation from the mean, 2.5 standard deviations from the mean, or 3 standard deviations from the mean. 3. Label the number line for a standard normal distribution curve. Lesson.4 l The Standard Normal Probability Distribution 797

46 4. Shade the area under the standard normal distribution curve that represents each percentage. Then, calculate each percentage. a. The percentage of the data values below 0 standard deviations. b. The percentage of the data values between 1 and 0 standard deviations. c. The percentage of the data values between 0 and 2 standard deviations. 798 Chapter l Distributions of Continuous Data

47 d. The percentage of the data values between 1 and 2 standard deviations. e. The percentage of the data values between 2 and 1 standard deviation. f. The percentage of the data values below 1 standard deviation. Lesson.4 l The Standard Normal Probability Distribution 799

48 g. The percentage of the data values above 2 standard deviations. Problem 3 Test It The standard normal distribution is often used to compare two different data sets measured on the same characteristic. For example, men are typically taller than women, but since adult height is normally distributed we can compare a man and woman s height relative to their gender. You can also compare how you performed on two different exams in terms of how you did in relation to the other students in your class, provided the exam scores are normally distributed. You will work more with the conversion to and from the standard normal distribution in the next lesson. 1. On a history test, the mean score for the class was 32 points with a standard deviation of 2 points. On a biology test, the mean score for the class was 63 points with a standard deviation of 7 points. The normal distribution for each test is drawn on the same number line. a. Describe the position of the mean for each test on the normal distribution. b. Which normal distribution curve would be wider and more spread out? Explain. c. Which normal distribution curve would be taller? Explain. 800 Chapter l Distributions of Continuous Data

49 2. A standard normal distribution for each test is shown. History Test Biology Test a. For each test, what score is represented by 0? Explain. Lesson.4 l The Standard Normal Probability Distribution 801

50 b. For each test, what score is represented by 1 and 1? Explain. c. For each test, what score is represented by 2 and 2? Explain. Be prepared to share your methods and solutions. 802 Chapter l Distributions of Continuous Data

51 .5 Catching Some Z s? Z-Scores and the Standard Normal Distribution Objectives In this lesson you will: l Calculate and interpret z-scores. l Apply properties of z-scores and the standard normal distribution. l Interpret the area below a z-score as a probability. Key Term l z-score Problem 1 Z-Z-Z s 1. A hybrid car has an average fuel efficiency of 54 miles per gallon (mpg) with a standard deviation of 6 mpg. Label the number line below the normal curve. Normal Curve Standard Normal Curve Lesson.5 l Z-Scores and the Standard Normal Distribution 803

52 2. Use the normal curve and the standard normal curve in Question 1 to answer each question. a. What does a value of 2 on the standard normal curve mean? What value on the normal curve is the same as a value of 2 on the standard normal curve? b. What value on the standard normal curve is the same as a value of 60 on the normal curve? What does each value mean? c. What value on the standard normal curve is the same as a value of 57 on the normal curve? Explain. d. What value on the standard normal curve is the same as a value of 43 on the normal curve? Explain. A z-score is a number that describes how many standard deviations from the mean a particular value is. The numbers on the number line of the standard normal curve are z-scores. The following formula can be used to calculate a z-score for a particular value. x z where z represents the z-score, x represents the particular data value, represents the mean, and represents the standard deviation. 804 Chapter l Distributions of Continuous Data

53 Calculating a z-score is said to standardize the particular data value because it can now be compared with different data sets measured on the same characteristic. 3. Check your answers for Question 2 using the z-score formula. a. What value on the normal curve is the same as a value of 2 on the standard normal curve? b. What value on the standard normal curve is the same as a value of 60 on the normal curve? What does each value mean? c. What value on the standard normal curve is the same as a value of 57 on the normal curve? Explain. d. What value on the standard normal curve is the same as a value of 43 on the normal curve? Explain. 4. A data set has a mean of 5 and standard deviation of 0.3. a. Calculate the z-score for a data value of 4. b. Calculate the z-score for a data value of Lesson.5 l Z-Scores and the Standard Normal Distribution 805

54 c. Calculate the data value with a z-score of d. Calculate the data value with a z-score of Shade each area on the standard normal curve. Then determine each area. a. Area between z 1 and z b. Area below z Chapter l Distributions of Continuous Data

55 c. Area between z 2 and z d. Area above z If a z-score is negative, what does that tell you about the value associated with it? 7. If a value is higher than the mean associated with it, what does that tell you about the z-score? Lesson.5 l Z-Scores and the Standard Normal Distribution 807

56 Problem 2 Z-Scores and Area 1. Consider the standard normal distribution shown a. Shade the area between z 0 and z 1. How wide is the interval? b. What is the area between z 0 and z 1? c. Shade the area between z 2 and z 1. How wide is the interval? d. What is the area between z 2 and z 1? e. Compare the width of the intervals and the area shaded. What do you notice? You can use the Empirical Rule for Normal Distributions to determine the area and probability of whole number z-scores. But what if the z-score is not a whole number? Two methods that can be used are a z-score table and a graphing calculator. 808 Chapter l Distributions of Continuous Data

57 A z-score table is provided at the end of this lesson and summarizes the area under the standard normal curve below each z-score. l The first column represents the ones and tenths place of the z-score. l The first row represents the hundredths place of the z-score. l Every other cell of the table represents the area under the standard normal curve below each z-score. To use a z-score table to determine the area under the standard normal curve below a z-score, locate the row and column that includes the ones, tenths, and hundredths place for the z-score. Then locate the number in the table where the row and column intersects. For example, locate the z-score 1.28 in the table and determine the area under the standard normal curve below z l For 1.28 the one s and tenth s place is 1.2 and the hundredths place is 8. l Read across the row containing 1.2 until you reach the column containing l The intersection of the row and column is the area. l The area below z 1.28 is or 89.97%. Z Examine both tables and note any patterns you see: Lesson.5 l Z-Scores and the Standard Normal Distribution 809

58 3. Why does the table only include z-scores between 3.49 and 3.49? 4. Why does a z-score of 0 have an area of ? 5. Determine the area under the standard normal curve below each z-score using the z-score table. Then, label the z-score on the number line and shade the area below the z-score. a. z 2.35 b. z 0.56 c. z 2.98 d. z 1.04 To determine the area under the standard normal curve below a z-score using a graphing calculator, perform the following steps. l Press the 2ND button and the VARS button to select the DISTR menu. l Select 2: normalcdf( l Enter the interval of z-scores, the mean of 0, and the standard deviation of 1 separated by commas. l Press ENTER. 810 Chapter l Distributions of Continuous Data

59 For example, calculate the area under the standard normal curve below a z-score of l The interval that represents the z-scores below 1.28 is to Most graphing calculators do not have a way to enter infinity. Instead of, specify the interval as a really large negative number. Enter the interval as 1E99 to Entering 1E99 is equivalent to a 1 followed by 99 zeroes, which is a really large negative number. l Enter normalcdf( 1E99, 1.28, 0, 1) and press ENTER. The value is which is the same value shown in the table for a z-score of Determine the area under the standard normal curve below each z-score using a graphing calculator. Compare each answer to the answer you determined in Question 5 using the z-score table. a. z 2.35 b. z 0.56 c. z 2.98 d. z 1.04 Problem 3 More Z s 1. Charles and Tamika are having an argument. Charles says that if a z-score is negative, then the value associated with it must also be negative. Tamika disagrees. Who is correct? Explain. 2. The mean of a data set is 83. Katrina calculated a positive z-score for a value of 73. Why is Katrina s answer incorrect? 3. When looking up the area associated with a z-score of 1.32, Tony got Why is Tony s answer incorrect? Lesson.5 l Z-Scores and the Standard Normal Distribution 811

60 4. Recall the hybrid car with an average fuel efficiency of 54 mpg and a standard deviation of 6 mpg. a. What is the z-score associated with 52 mpg? b. What percentage of hybrid cars get less than 52 mpg? c. What percentage of hybrid cars get more than 52 mpg? d. What is the probability that a randomly selected hybrid car gets less than 40 mpg? e. What is the probability that a randomly selected hybrid car gets less than 60 mpg? Be prepared to share your methods and solutions. 812 Chapter l Distributions of Continuous Data

61 z-scores and area below the z-score z Lesson.5 l Z-Scores and the Standard Normal Distribution 813

62 z Chapter l Distributions of Continuous Data

63 .6 Above and In-Between Probabilities Above and Between Z-Scores Objectives In this lesson you will: l Calculate the area above a z-score using a z-score table and a graphing calculator. l Calculate the area between two z-scores using a z-score table and a graphing calculator. Problem 1 Above Z-Scores Recall the hybrid car with an average fuel efficiency of 54 mpg and a standard deviation of 6 mpg. 1. What percentage of hybrid cars gets less than 50 mpg? 2. Shade the area of the standard normal curve that corresponds to the percentage of hybrid cars that get less than 50 mpg. What is the area of the shaded region? Lesson.6 l Probabilities Above and Between Z-Scores 815

64 3. How much area is under the standard normal curve? 4. What is the area of the unshaded region? How do you know? 5. What does the unshaded region represent? 6. If the area of a shaded region under the curve is known, how can you calculate the area of the unshaded region under the curve? 7. You know the mean and standard deviation of a data set. How can you use a z-score table to calculate the percentage of data values below a given data value? 8. You know the mean and standard deviation of a data set. How can you use a z-score table to calculate the percentage of data values above a given data value? 816 Chapter l Distributions of Continuous Data

65 9. Determine the area under the standard normal curve above each z-score using the z-score table. Then, label the z-score on the number line and shade the area above the z-score. a. z 2.35 b. z 0.56 c. z 0.67 d. z 2.32 To determine the area under the standard normal curve above a z-score using a graphing calculator, perform the following steps. l Press the 2ND button and the VARS button to select the DISTR menu. l Select 2: normalcdf(. l Enter the interval of z-scores, the mean of 0, and the standard deviation of 1 separated by commas. l Press ENTER. For example, calculate the area under the standard normal curve above a z-score of l The interval that represents the z-scores above 1.28 is 1.28 to. Most graphing calculators do not have a way to enter infinity. Instead of, specify the interval as a really large positive number. Enter the interval as 1.28 to 1E99. Entering 1E99 is equivalent to a 1 followed by 99 zeroes, which is a really large positive number. l Enter normalcdf(1.28, 1E99, 0, 1) and press ENTER. The value is which is the same value shown in the table for a z-score of Lesson.6 l Probabilities Above and Between Z-Scores 8

66 10. Determine the area under the standard normal curve above each z-score using a graphing calculator. Compare each answer to the answer you determined in Question 9 using the z-score table. a. z 2.35 b. z 0.56 c. z 0.67 d. z 2.32 Problem 2 In-Between Z-Scores 1. Suppose we wanted to know the percentage of hybrid cars that get between 50 mpg and 60 mpg. Write an inequality expressing this interval. 2. Shade the area under the normal curve that represents the percentage of cars that get between 50 mpg and 60 mpg. 3. How could you determine the percentage of hybrid cars that get between 50 mpg and 60 mpg? 4. Examine the normal curves shown that represent the fuel efficiency of hybrid cars. Describe what each shaded region represents Chapter l Distributions of Continuous Data

67 5. Both graphs from Question 4 are shown on the same number line. Shade the area under the normal curve that represents the percentage of cars that get between 50 mpg and 60 mpg Describe the area that is not part of the solution. How could you use that area to determine the area of interest? 7. You know the mean and standard deviation of a data set. How can you use a z-score table to calculate the percentage of data values between two given data values? Lesson.6 l Probabilities Above and Between Z-Scores 819

68 8. Determine the area under the standard normal curve between each pair of z-scores using the z-score table. a. between z 1.00 and z 2.00 b. between z 1.53 and z 0.52 c. between z 2.1 and z You know the mean and standard deviation of a data set. How can you use a graphing calculator to calculate the percentage of data values between two given data values? Provide an example in your explanation. 820 Chapter l Distributions of Continuous Data

69 10. Determine the area under the standard normal curve between each pair of z-scores using a graphing calculator. Compare each answer to the answer you determined in Question 8 using the z-score table. a. between z 1.00 and z 2.00 b. between z 1.53 and z 0.52 c. between z 2.1 and z 0.57 Problem 3 More Grasshoppers Recall the hybrid car with an average fuel efficiency of 54 mpg and a standard deviation of 6 mpg. 1. What percentage of hybrid cars get more than 60 mpg? 2. What is the probability that a randomly selected hybrid car gets between 55 mpg and 60 mpg? 3. What is the probability that a randomly selected hybrid car gets more than 52 mpg? Lesson.6 l Probabilities Above and Between Z-Scores 821