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  z Score (or standard score) the number of standard deviations that a given value x is above or below the mean Measures of Position (Section 2.6)

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

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

5 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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 5-5The Central Limit Theorem 5-6 Normal Distribution as Approximation to Binomial Distribution (skipped) 5-7Determining Normality

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

7 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 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.

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

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

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

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

12 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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? 1 34% 0z

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

14 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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) Finding Probabilities from Standard(Z) scores

15 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. Since  = 0 and  =1, the values are standard or z-scores.

16 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. Since  = 0 and  =1, the values are standard or z-scores.

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

18 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

19 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman P ( 0 < x < 1.58 ) = Normalcdf(0, 1.58) =.443 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.

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

21 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman There is 44.3% of the thermometers with readings between 0 and 1.58 degrees Area = P ( 0 < x < 1.58 ) = 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.

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 degrees and 0 degrees

23 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 Area = P ( < x < 0 ) = Normalcdf(-2.43, 0)= NOTE: Although a z score can be negative, the area under the curve (or the corresponding probability) can never be negative.

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

25 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 0.5 Remember the graph represents the entire distribution with its area of 1.

26 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 = P(z>1.27) = normalcdf(1.27, 9999) = z = 1.27

27 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 = z = 1.20 Area =.104 z = 2.30 P(1.20 <z< 2.30) = normalcdf(1.20, 2.30) = 0.104

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

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

30 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 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 z- score(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 invNorm() function in your calculator. z = invNorm(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.

31 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman ( z score will be positive ) z 95% Finding z Scores when Given Probabilities Find the z-score that denotes the 95th Percentile. z = invNorm(.95) =1.65

32 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman z Find the z-score that separate the bottom 10% and upper 90% % Bottom 10% 10% ( z score will be negative) Finding z Scores when Given Probabilities z = invNorm(.10) = -1.28

33 Chapter 5. Section 5-1 and 5-2. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 0 60% 0.60 z = invNorm(.20) = Find the z-scores that denotes the middle 60% Finding z Scores when Given Probabilities 0.20 Bottom 20% 20% % z = invNorm(.80) = 0.84