ELEMENTARY Chapter 5 STATISTICS Normal Probability Distributions MARIO

ELEMENTARY Chapter 5 STATISTICS Normal Probability Distributions MARIO F. TRIOLA EIGHTH Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION 1

Measures of Position (Section 2. 6) v z Score (or standard score) the number of standard deviations that a given value x is above or below the mean Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2

Measures of Position z score Sample x x z= s Population x µ z= Round to 2 decimal places Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3

FIGURE 2 -16 Interpreting Z Scores Unusual Values Ordinary Values -3 -2 -1 0 Unusual Values 1 2 3 Z Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4

Chapter 5 Normal Probability Distributions 5 -1 Overview 5 -2 The Standard Normal Distribution 5 -3 Normal Distributions: Finding Probabilities 5 -4 Normal Distributions: Finding Values 5 -5 The Central Limit Theorem 5 -6 Normal Distribution as Approximation to Binomial Distribution (skipped) 5 -7 Determining Normality Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5

Normal Distribution v Continuous random variable v Graph is symmetric and bell shaped v Area under the curve is equal to 1, therefore there is correspondence between area and probability. µ Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6

Heights of Adult Men and Women • Normal distributions (bell shaped) are a family of distributions that have the same general shape. • The mean, mu, controls the center • Standard deviation, sigma, controls the spread. Women: µ = 63. 6 = 2. 5 Figure 5 -4 Men: µ = 69. 0 = 2. 8 63. 6 69. 0 Height (inches) Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7

5 -2 The Standard Normal Distribution Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8

Definition Standard Normal Distribution a normal probability distribution that has a mean of 0 and a standard deviation of 1 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9

The Empirical Rule Standard Normal Distribution: µ = 0 and = 1 99. 7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation 34% 2. 4% 0. 1% 13. 5% -3 -2 13. 5% -1 0 1 2 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3 z 10

Example: Using your knowledge of the Emperical rule, what is the probability that a value falls between the mean and 1 standard deviation above the mean? Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11

Example: Using your knowledge of the Emperical rule, what is the probability that a value falls between the mean and 1 standard deviation above the mean? 34% 0 1 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman z 12

What if we need to find the probability of a value that does not fall on a standard deviation? Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13

Finding Probabilities from Standard(Z) scores Step 1: Draw a normal curve with the centerline labeled 0 on the x-axis. Step 2: Label given value(s) on the appropriate location on the x-axis. Draw vertical lines on the normal curve above the given values. Step 3: Shade the area of the region in question. Step 4: Use calculator function normalcdf() to find the area (probability) in question. normalcdf(left boundary, right boundary) Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1. 58 degrees. Since = 0 and =1, the values are standard or z-scores. Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1. 58 degrees. Since = 0 and =1, the values are standard or z-scores. 0 1. 58 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1. 58 degrees. Since = 0 and =1, the values are standard or z-scores. P ( 0 < x < 1. 58 ) = ? 0 1. 58 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17

Table A-2 Standard Normal (z) Distribution z . 00 . 01 . 02 . 03 . 04 . 05 . 06 . 07 . 08 . 09 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1. 0 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 2. 0 2. 1 2. 2 2. 3 2. 4 2. 5 2. 6 2. 7 2. 8 2. 9 3. 0 . 0000. 0398. 0793. 1179. 1554. 1915. 2257. 2580. 2881. 3159. 3413. 3643. 3849. 4032. 4192. 4332. 4452. 4554. 4641. 4713. 4772. 4821. 4861. 4893. 4918. 4938. 4953. 4965. 4974. 4981. 4987 . 0040. 0438. 0832. 1217. 1591. 1950. 2291. 2611. 2910. 3186. 3438. 3665. 3869. 4049. 4207. 4345. 4463. 4564. 4649. 4719. 4778. 4826. 4864. 4896. 4920. 4940. 4955. 4966. 4975. 4982. 4987 . 0080. 0478. 0871. 1255. 1628. 1985. 2324. 2642. 2939. 3212. 3461. 3686. 3888. 4066. 4222. 4357. 4474. 4573. 4656. 4726. 4783. 4830. 4868. 4898. 4922. 4941. 4956. 4967. 4976. 4982. 4987 . 0120. 0517. 0910. 1293. 1664. 2019. 2357. 2673. 2967. 3238. 3485. 3708. 3907. 4082. 4236. 4370. 4484. 4582. 4664. 4732. 4788. 4834. 4871. 4901. 4925. 4943. 4957. 4968. 4977. 4983. 4988 . 0160. 0557. 0948. 1331. 1700. 2054. 2389. 2704. 2995. 3264. 3508. 3729. 3925. 4099. 4251. 4382. 4495. 4591. 4671. 4738. 4793. 4838. 4875. 4904. 4927. 4945. 4959. 4969. 4977. 4984. 4988 . 0199. 0596. 0987. 1368. 1736. 2088. 2422. 2734. 3023. 3289. 3531. 3749. 3944. 4115. 4265. 4394. 4505. 4599. 4678. 4744. 4798. 4842. 4878. 4906. 4929. 4946. 4960. 4978. 4984. 4989 . 0239. 0636. 1026. 1406. 1772. 2123. 2454. 2764. 3051. 3315. 3554. 3770. 3962. 4131. 4279. 4406. 4515. 4608. 4686. 4750. 4803. 4846. 4881. 4909. 4931. 4948. 4961. 4979. 4985. 4989 . 0279. 0675. 1064. 1443. 1808. 2157. 2486. 2794. 3078. 3340. 3577. 3790. 3980. 4147. 4292. 4418. 4525. 4616. 4693. 4756. 4808. 4850. 4884. 4911. 4932. 4949. 4962. 4979. 4985. 4989 . 0319. 0714. 1103. 1480. 1844. 2190. 2517. 2823. 3106. 3365. 3599. 3810. 3997. 4162. 4306. 4429. 4535. 4625. 4699. 4761. 4812. 4854. 4887. 4913. 4934. 4951. 4963. 4973. 4980. 4986. 4990 . 0359. 0753. 1141. 1517. 1879. 2224. 2549. 2852. 3133. 3389. 3621. 3830. 4015. 4177. 4319. 4441. 4545. 4633. 4706. 4767. 4817. 4857. 4890. 4916. 4936. 4952. 4964. 4974. 4981. 4986. 4990 * * Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1. 58 degrees. Since = 0 and =1, the values are standard or z-scores. P ( 0 < x < 1. 58 ) = Normalcdf(0, 1. 58) =. 443 0 1. 58 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1. 58 degrees. Since = 0 and =1, the values are standard or z-scores. Area = 0. 443 P ( 0 < x < 1. 58 ) = 0. 443 0 1. 58 The probability that the chosen thermometer will measure freezing water between 0 and 1. 58 degrees is 0. 443. Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 20

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water and if one thermometer is randomly selected, find the probability that it reads freezing water between 0 degrees and 1. 58 degrees. Since = 0 and =1, the values are standard or z-scores. Area = 0. 443 P ( 0 < x < 1. 58 ) = 0. 443 0 1. 58 There is 44. 3% of thermometers with readings between 0 and 1. 58 degrees. Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 21

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water, and if one thermometer is randomly selected, find the probability that it reads freezing water between -2. 43 degrees and 0 degrees. -2. 43 0 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 22

Example: If thermometers have an average (mean) reading of 0 degrees and a standard deviation of 1 degree for freezing water, and if one thermometer is randomly selected, find the probability that it reads freezing water between -2. 43 degrees and 0 degrees. Area = 0. 493 -2. 43 P ( -2. 43 < x < 0 ) = Normalcdf(-2. 43, 0)= 0. 493 0 NOTE: Although a z score can be negative, the area under the curve (or the corresponding probability) can never be negative. Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23

Notation P(a < z < b) → Normalcdf(a, b) denotes the probability that the z score is between a and b P(z > a) → Normalcdf(a, 9999) denotes the probability that the z score is greater than a P (z < a) → Normalcdf(-9999, a) denotes the probability that the z score is less than a Note: 9999 is used to represent infinity Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 24

Probability of Half of a Distribution Remember the graph represents the entire distribution with its area of 1. 0. 5 =0 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 25

Finding the Area to the Right of z = 1. 27 P(z>1. 27) = normalcdf(1. 27, 9999) = 0. 1020 0 z = 1. 27 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 26

Finding the Area Between z = 1. 20 and z = 2. 30 P(1. 20 <z< 2. 30) = normalcdf(1. 20, 2. 30) = 0. 104 Area =. 104 0 z = 1. 20 z = 2. 30 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 27

Interpreting Area Correctly Figure 5 -10 ‘greater than ‘at least x’ x’ ‘more than x’ ‘not less than x’ x Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman x 28

Interpreting Area Correctly x’ ‘greater than ‘at least x’ ‘more than x’ ‘not less than x’ x ‘less than ‘at most x x’ x’ ‘no more than x’ ‘not greater than x’ x Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman x 29

Finding a z - score when given a probability 1. Draw a bell-shaped curve, draw the centerline, and shade the region under the curve that corresponds to the given probability. The line(s) that bound the region denotes the zscore(s) you are trying to find. 2. Using the probability representing the area bounded by the z -score, convert it to area to the left of the z-score and enter that value into inv. Norm() function in your calculator. z = inv. Norm(area to the left) This command finds the z-score that has area to the left of the boundary under the normal curve. Since the area represent probability the argument must be between 0 and 1 inclusive. 3. If the z score is positioned to the left of the centerline, make sure it’s negative. Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 30

Finding z Scores when Given Probabilities Find the z-score that denotes the 95 th Percentile. 95% 0. 95 0 z = inv. Norm(. 95) =1. 65 z ( z score will be positive ) Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 31

Finding z Scores when Given Probabilities Find the z-score that separate the bottom 10% and upper 90% 10% Bottom 10% 0. 90 0. 10 z (z score will be negative) 0 z = inv. Norm(. 10) = -1. 28 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 32

Finding z Scores when Given Probabilities Find the z-scores that denotes the middle 60% 20% 60% Bottom 20% 0. 60 0. 20 z = inv. Norm(. 20) = -0. 84 0. 20 0 z = inv. Norm(. 80) = 0. 84 Chapter 5. Section 5 -1 and 5 -2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 33