Lesson Z-Scores and Normal Distributions

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STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION 1 A large company is hiring one employee for a top position. Your team will recommend who gets the job.

1 STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION 1 A large company is hiring one employee for a top position. Your team will recommend who gets the job. After completing many interviews, reference checks, and verification of written applications, you narrowed the choice down to two outstanding candidates. Both are well regarded and equally qualified. Since the job can be very stressful and requires a significant amount of travel, the final component in the hiring process requires each to take a personality test. Both candidates have the results of such a test in their file. Unfortunately, they did not take the same test. Using the following results, whom do you recommend for the position? Be prepared to justify your choice to the company chief executive officer. Note: The Gallup Organization actually administers several types of tests to aid companies in the hiring process. They have been a key part of the hiring process for corporate leaders, teachers, and NBA basketball players. Candidate A Test taken: Stress Sanity Simulator Mean score: 742 Standard deviation: 28 Candidate score: 801 Candidate B Test taken: Modeling Mind Management Mean score: 67 Standard deviation: 12 Candidate score: 94

2 STATWAY STUDENT HANDOUT 2 The Normal Curve In a previous lesson, you explored continuous uniform distributions and probabilities. Few real-life distributions are uniform. There are, however, many distributions that are approximately normal. YOU NEED TO KNOW The Normal Distribution Normal distributions follow the following guidelines: The mean, median, and mode are equal The normal curve is bell-shaped and symmetric about the mean The area under the curve is 1 (This will be helpful in finding probabilities) The normal curve approaches, but never touches the x-axis as it extends farther and farther from the mean. You can describe a normal distribution by its mean and standard deviation. It could have a mean of 23 and a standard deviation of 1 (curve C), or a mean of 15 and a standard deviation of 1.5 (curve A). A B C D

3 STATWAY STUDENT HANDOUT 3 TRY THESE Match the following graphs of normal distributions with the appropriate mean and standard deviation. A B C D Mean 15 and standard deviation 4 3 Mean 13 and standard deviation 2 4 Mean 15 and standard deviation 1 5 Mean 17 and standard deviation 2

4 STATWAY STUDENT HANDOUT 4 NEXT STEPS And just like uniform distributions, when you let the area under the curve equal 1, you can use the area to find probabilities. Finding the area under the normal curve requires more difficult mathematics than finding the area of a rectangle. Fortunately, all normal distributions have the same relationship between area and the number of standard deviations from the mean. 68% 95% 99.7%

5 STATWAY STUDENT HANDOUT 5 YOU NEED TO KNOW The Empirical Rule ( Rule) In a normal distribution 68% of the values lie within 1 standard deviation of the mean, 95% of the values lie within 2 standard deviations of the mean*, and 99.7% of the values lie within 3 standard deviations of the mean. 99.7% Range 95% Range 68% Range *Note: Values that are more than 2 standard deviations from the mean are considered to be unusual.

6 STATWAY STUDENT HANDOUT 6 You can use Empirical Rule to find probabilities (or the proportion of data) that lies between 2 points. Suppose an approximately normal distribution has mean 25 and standard deviation 2. We want to be able to answer the question What proportion of the data is between 23 and 27 units? In order to do this, we start with a diagram of the normal curve. Identify the mean on the x- axis (25). Mark off lengths of 2 (the standard deviation), starting with the mean as the center Since 23 and 27 are 1 standard deviation above and below the mean, the shaded area is 0.68.

7 STATWAY STUDENT HANDOUT 7 TRY THESE 6 The speed limit on a stretch of country road is 55 miles per hour (mph). A device used to measure the actual speed of cars traveling on the road finds the distribution of speeds to be approximately normal with a mean of 55 and a standard deviation of 5. Label the normal curve with the appropriate scale. Then find the probabilities using the Empirical Rule. A What is the probability that a randomly selected car on this road is traveling between 45 and 65 mph? B What is the probability that a randomly selected car on this road is traveling between 55 and 60 mph? C What is the probability that a randomly selected car on this road is traveling above 50 mph? D What is the probability that a randomly selected car on this road is traveling below 45 mph? E What is the probability that a randomly selected car on this road is traveling above 55 mph? F What is the probability that a randomly selected car on this road is traveling between 50 and 65 mph?

8 STATWAY STUDENT HANDOUT 8 TAKE IT HOME 1 The Westwood Warrior basketball team is trying to improve its offensive strategies. Some opponents primarily use a zone defense, while others primarily use a man-to-man defense. On average, the Warriors score 67 points against teams with a zone defense and 62 points against teams with a man-to-man defense. The standard deviations are 8 and 5, respectively. Since the improved offensive strategies have been put into practice, the Warriors have played two games with the following results. McNeil Mavericks Maverick defense: zone Warrior points: 77 Round Rock Dragons Dragon defense: man-to-man Warrior points: 71 Which offensive strategy zone or man-to-man appears to have improved the most? (Of course, there are many factors involved in playing basketball. For your answer, assume the difference is due to the new offensive strategies.)

9 STATWAY STUDENT HANDOUT 9 2 The average height of an adult male can be reasonably described by a normal distribution with a mean of 69 inches and a standard deviation of 2.5 inches. Find the probability of randomly selecting a male in the following categories using the Empirical Rule. For each problem, sketch the curve with the appropriate scale and shade the area described. A What is the probability that a randomly selected male is more than 74 inches tall? B What is the probability that a randomly selected male is between 64 and 69 inches tall? C What is the probability that a randomly selected male is less than 61.5 inches tall? D What is the probability that a randomly selected male is between 66.5 and 76.5 inches tall?

10 STATWAY STUDENT HANDOUT This lesson is part of STATWAY, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel STATWAY and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.

Study Guide and Intervention

Study Guide and Intervention Normal and Skewed Distributions A continuous probability distribution is represented by a curve. Types of Continuous Distributions Normal Positively Skewed Negatively Skewed