1 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Chapter 5 Normal Probability Distributions

2 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  Continuous random variable  Normal distribution Overview 5-1

3 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  Continuous random variable  Normal distribution Curve is bell shaped and symmetric µ Score Overview 5-1 Figure 5-1

4 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  Continuous random variable  Normal distribution Curve is bell shaped and symmetric µ Score Formula 5-1 Overview 5-1 Figure 5-1 x - µ 2 y = 1 2  e  2  ( )

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

6 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  Uniform Distribution a probability distribution in which the continuous random variable values are spread evenly over the range of possibilities; the graph results in a rectangular shape. Definitions

7 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  Density Curve (or probability density function) the graph of a continuous probability distribution Definitions

8 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  Density Curve (or probability density function) :  The graph of a continuous probability distribution Definitions 1. The total area under the curve must equal Every point on the curve must have a vertical height that is 0 or greater.

9 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.

10 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Times in First or Last Half Hours Figure 5-3

11 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Heights of Adult Men and Women Women: µ = 63.6  = 2.5 Men: µ = 69.0  = Height (inches) Figure 5-4

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

13 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Definition Standard Normal Deviation a normal probability distribution that has a mean of 0 and a standard deviation of z = 1.58 Figure 5-5 Figure 5-6 Area = Area found in Table A Score (z )

14 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Table A-2 Standard Normal Distribution µ = 0  = 1 0 x z

15 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman * * z Table A-2 Standard Normal ( z ) Distribution

16 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman To find: z Score the distance along horizontal scale of the standard normal distribution; refer to the leftmost column and top row of Table A-2 Area the region under the curve; refer to the values in the body of Table A-2

17 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.

18 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 P ( 0 < x < 1.58 ) =

19 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman * * z Table A-2 Standard Normal ( z ) Distribution

20 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 Area = P ( 0 < x < 1.58 ) =

21 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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. The probability that the chosen thermometer will measure freezing water between 0 and 1.58 degrees is Area = P ( 0 < x < 1.58 ) =

22 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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. There is 44.29% of the thermometers with readings between 0 and 1.58 degrees Area = P ( 0 < x < 1.58 ) =

23 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Using Symmetry to Find the Area to the Left of the Mean NOTE: Although a z score can be negative, the area under the curve (or the corresponding probability) can never be negative. (a) (b) Because of symmetry, these areas are equal. Equal distance away from z = 2.43 z = Figure 5-7

24 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 degrees and 0 degrees. The probability that the chosen thermometer will measure freezing water between and 0 degrees is Area = P ( < x < 0 ) =

25 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

26 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - s x x + sx + s 68% within 1 standard deviation 34% The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

27 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - 2s x - s x x + 2s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 13.5% The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

28 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - 3s x - 2s x - s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean 0.1% 2.4% 13.5% The Empirical Rule Standard Normal Distribution: µ = 0 and  = 1

29 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Probability of Half of a Distribution 0 0.5

30 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding the Area to the Right of z = Value found in Table A-2 This area is = z = 1.27 Figure 5-8

31 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding the Area Between z = 1.20 and z = (from Table A-2 with z = 2.30) Area A is = z = 1.20 A z = 2.30 Figure 5-9

32 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P(a < z < b) denotes the probability that the z score is between a and b P( z > a) denotes the probability that the z score is greater than a P ( z < a) denotes the probability that the z score is less than a Notation

33 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 5-10 Interpreting Area Correctly Add to 0.5 x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x Subtract from 0.5

34 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 5-10 Interpreting Area Correctly Add to 0.5 x Add to 0.5 x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x Subtract from 0.5 x Subtract from 0.5 ‘less than x ’ ‘at most x ’ ‘no more than x ’ ‘not greater than x ’

35 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 5-10 Interpreting Area Correctly Add to 0.5 x Add to 0.5 x ‘greater than x ’ ‘at least x ’ ‘more than x ’ ‘not less than x ’ x Subtract from 0.5 x Subtract from 0.5 x1x1 x2x2 Add ‘less than x ’ ‘at most x ’ ‘no more than x ’ ‘not greater than x ’ ‘between x 1 and x 2 ’ AB Use A = C - B x1x1 x2x2 C

36 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Finding a z - score when given a probability Using Table A-2 1. Draw a bell-shaped curve, draw the centerline, and identify the region under the curve that corresponds to the given probability. If that region is not bounded by the centerline, work with a known region that is bounded by the centerline. 2.Using the probability representing the area bounded by the centerline, locate the closest probability in the body of Table A-2 and identify the corresponding z score. 3.If the z score is positioned to the left of the centerline, make it a negative.

37 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman z %5% 5% or 0.05 Finding z Scores when Given Probabilities FIGURE 5-11 Finding the 95th Percentile ( z score will be positive )

38 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman %5% 5% or 0.05 Finding z Scores when Given Probabilities FIGURE 5-11 Finding the 95th Percentile ( z score will be positive)

39 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman z % FIGURE 5-12 Finding the 10th Percentile Bottom 10% 10% ( z score will be negative) Finding z Scores when Given Probabilities

40 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman % FIGURE 5-12 Finding the 10th Percentile Bottom 10% 10% Finding z Scores when Given Probabilities ( z score will be negative)

41 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Assignment Page 240: 1-36 all