Transcript Chapter7
Chapter 7 The Normal Probability Distribution Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section 7.1 Properties of the Normal Distribution Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1. 2. 3. 4. 5-3 Use the uniform probability distribution Graph a normal curve State the properties of the normal curve Explain the role of area in the normal density function Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 • Use the Uniform Probability Distribution 5-4 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Illustrating the Uniform Distribution Suppose that United Parcel Service is supposed to deliver a package to your front door and the arrival time is somewhere between 10 am and 11 am. Let the random variable X represent the time from 10 am when the delivery is supposed to take place. The delivery could be at 10 am (x = 0) or at 11 am (x = 60) with all 1-minute interval of times between x = 0 and x = 60 equally likely. That is to say your package is just as likely to arrive between 10:15 and 10:16 as it is to arrive between 10:40 and 10:41. The random variable X can be any value in the interval from 0 to 60, that is, 0 < X < 60. Because any two intervals of equal length between 0 and 60, inclusive, are equally likely, the random variable X is said to follow a uniform probability distribution. 7-5 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A probability density function (pdf) is an equation used to compute probabilities of continuous random variables. It must satisfy the following two properties: 1.The total area under the graph of the equation over all possible values of the random variable must equal 1. 2.The height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable. 5-6 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The graph below illustrates the properties for the “time” example. Notice the area of the rectangle is one and the graph is greater than or equal to zero for all x between 0 and 60, inclusive. Because the area of a rectangle is height times width, and the width of the rectangle is 60, the height must be 1/60. 7-7 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Values of the random variable X less than 0 or greater than 60 are impossible, thus the function value must be zero for X less than 0 or greater than 60. 7-8 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The area under the graph of the density function over an interval represents the probability of observing a value of the random variable in that interval. 7-9 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Area as a Probability The probability of choosing a time that is between 15 and 30 minutes after the hour is the area under the uniform density function. Area = P(15 < x < 30) = 15/60 = 0.25 15 7-10 30 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 • Graph a Normal Curve 5-11 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Relative frequency histograms that are symmetric and bell-shaped are said to have the shape of a normal curve. 7-12 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. If a continuous random variable is normally distributed, or has a normal probability distribution, then a relative frequency histogram of the random variable has the shape of a normal curve (bell-shaped and symmetric). 7-13 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 7-14 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 3 • State the Properties of the Normal Curve 5-15 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Properties of the Normal Density Curve 1. It is symmetric about its mean, μ. 2. Because mean = median = mode, the curve has a single peak and the highest point occurs at x = μ. 3.It has inflection points at μ – σ and μ – σ 4.The area under the curve is 1. 5.The area under the curve to the right of μ equals the area under the curve to the left of μ, which equals 1/2. 5-16 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 6. As x increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis. As x decreases without bound (gets more and more negative), the graph approaches, but never reaches, the horizontal axis. 7. The Empirical Rule: Approximately 68% of the area under the normal curve is between x = μ – σ and x = μ + σ; approximately 95% of the area is between x = μ – 2σ and x = μ + 2σ; approximately 99.7% of the area is between x = μ – 3σ and x = μ + 3σ. 5-17 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 7-18 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 4 • Explain the Role of Area in the Normal Density Function 5-19 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE A Normal Random Variable The data on the next slide represent the heights (in inches) of a random sample of 50 two-year old males. (a) Draw a histogram of the data using a lower class limit of the first class equal to 31.5 and a class width of 1. (b) Do you think that the variable “height of 2year old males” is normally distributed? 7-20 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 36.0 34.7 34.4 33.2 35.1 38.3 37.2 7-21 36.2 33.4 35.7 36.1 35.2 33.6 39.3 34.8 37.4 37.9 35.2 34.4 39.8 36.0 38.2 39.3 35.6 36.7 37.0 34.6 31.5 34.0 33.0 36.0 37.2 38.4 37.7 36.9 36.8 36.0 34.8 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 35.4 36.9 35.1 33.5 35.7 35.7 36.8 34.0 37.0 35.0 35.7 38.9 7-22 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. In the next slide, we have a normal density curve drawn over the histogram. How does the area of the rectangle corresponding to a height between 34.5 and 35.5 inches relate to the area under the curve between these two heights? 7-23 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 7-24 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 7-25 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Area under a Normal Curve Suppose that a random variable X is normally distributed with mean μ and standard deviation σ. The area under the normal curve for any interval of values of the random variable X represents either • the proportion of the population with the characteristic described by the interval of values or • the probability that a randomly selected individual from the population will have the characteristic described by the interval of values. 5-26 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Interpreting the Area Under a Normal Curve The weights of giraffes are approximately normally distributed with mean μ = 2200 pounds and standard deviation σ = 200 pounds. (a)Draw a normal curve with the parameters labeled. (b)Shade the area under the normal curve to the left of x = 2100 pounds. (c)Suppose that the area under the normal curve to the left of x = 2100 pounds is 0.3085. Provide two interpretations of this result. 7-27 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Interpreting the Area Under a Normal Curve μ = 2200 pounds and σ = 200 pounds (a), (b) (c) •The proportion of giraffes whose weight is less than 2100 pounds is 0.3085 •The probability that a randomly selected giraffe weighs less than 2100 pounds is 0.3085. 7-28 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section 7.2 Applications of the Normal Distribution Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1. Find and interpret the area under a normal curve 2. Find the value of a normal random variable 5-30 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 • Find and Interpret the Area Under a Normal Curve 5-31 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Standardizing a Normal Random Variable Suppose that the random variable X is normally distributed with mean μ and standard deviation σ. Then the random variable Z X is normally distributed with mean μ = 0 and standard deviation σ = 1.The random variable Z is said to have the standard normal distribution. 5-32 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Standard Normal Curve 7-33 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The table gives the area under the standard normal curve for values to the left of a specified Z-score, z, as shown in the figure. 7-34 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. IQ scores can be modeled by a normal distribution with μ = 100 and σ = 15. An individual whose IQ score is 120, is 1.33 standard deviations above the mean. z 7-35 x 120 100 1.33 15 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The area under the standard normal curve to the left of z = 1.33 is 0.9082. (Table V) 7-36 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Use the Complement Rule to find the area to the right of z = 1.33. 7-37 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Areas Under the Standard Normal Curve 7-38 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Finding the Area Under the Standard Normal Curve Find the area under the standard normal curve to the left of z = –0.38. Area to the left of z = –0.38 is 0.3520. 7-39 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Area under the normal curve to the right of zo = 1 – Area to the left of zo 7-40 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Finding the Area Under the Standard Normal Curve Find the area under the standard normal curve to the right of z = 1.25. Area right of 1.25 = 1 – area left of 1.25 = 1 – 0.8944 = 0.1056 7-41 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Finding the Area Under the Standard Normal Curve Find the area under the standard normal curve between z = –1.02 and z = 2.94. Area between –1.02 and 2.94 = (Area left of z = 2.94) – (area left of z = –1.02) = 0.9984 – 0.1539 = 0.8445 7-42 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Problem: Find the area to the left of x. Approach: Shade the area to the left of x. Solution: • Convert the value of x to a z-score. Use Table V to find the row and column that correspond to z. The area to the left of x is the value where the row and column intersect. • Use technology to find the area. 7-43 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Problem: Find the area to the right of x. Approach: Shade the area to the right of x. Solution: • Convert the value of x to a z-score. Use Table V to find the area to the left of z (also is the area to the left of x). The area to the right of z (also x) is 1 minus the area to the left of z. • Use technology to find the area. 7-44 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Problem: Find the area between x1 and x2. Approach: Shade the area between x1 and x2. Solution: • Convert the values of x to a z-scores. Use Table V to find the area to the left of z1 and to the left of z2. The area between z1 and z2 is (area to the left of z2) – (area to the left of z1). • Use technology to find the area. 7-45 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 • Find the Value of a Normal Random Variable 5-46 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Procedure for Finding the Value of a Normal Random Variable Step 1: Draw a normal curve and shade the area corresponding to the proportion, probability, or percentile. Step 2: Use Table V to find the z-score that corresponds to the shaded area. Step 3: Obtain the normal value from the formula x = μ + zσ. 5-47 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Finding the Value of a Normal Random Variable The combined (verbal + quantitative reasoning) score on the GRE is normally distributed with mean 1049 and standard deviation 189. (Source: http://www.ets.org/Media/Tests/GRE/pdf/994994.pdf.) What is the score of a student whose percentile rank is at the 85th percentile? 7-48 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Finding the Value of a Normal Random Variable The z-score that corresponds to the 85th percentile is the z-score such that the area under the standard normal curve to the left is 0.85. This z-score is 1.04. x = µ + zσ = 1049 + 1.04(189) = 1246 Interpretation: A person who scores 1246 on the GRE would rank in the 85th percentile. 7-49 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Finding the Value of a Normal Random Variable It is known that the length of a certain steel rod is normally distributed with a mean of 100 cm and a standard deviation of 0.45 cm. Suppose the manufacturer wants to accept 90% of all rods manufactured. Determine the length of rods that make up the middle 90% of all steel rods manufactured. 7-50 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Area = 0.05 Finding the Value of a Normal Random Variable Area = 0.05 z1 = –1.645 and z2 = 1.645 x1 = µ + z1σ = 100 + (–1.645)(0.45) = 99.26 cm x2 = µ + z2σ = 100 + (1.645)(0.45) = 100.74 cm Interpretation: The length of steel rods that make up the middle 90% of all steel rods manufactured would have lengths between 99.26 cm and 100.74 cm. 7-51 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The notation zα (pronounced “z sub alpha”) is the z-score such that the area under the standard normal curve to the right of zα is α. 5-52 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Section 7.3 Assessing Normality Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1. Use normal probability to assess normality 5-54 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Suppose that we obtain a simple random sample from a population whose distribution is unknown. Many of the statistical tests that we perform on small data sets (sample size less than 30) require that the population from which the sample is drawn be normally distributed. Up to this point, we have said that a random variable X is normally distributed, or at least approximately normal, provided the histogram of the data is symmetric and bell-shaped. This method works well for large data sets, but the shape of a histogram drawn from a small sample of observations does not always accurately represent the shape of the population. For this reason, we need additional methods for assessing the normality of a random variable X when we are looking at sample data. 7-55 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 • Use Normal Probability Plots to Assess Normality 5-56 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. A normal probability plot plots observed data versus normal scores. A normal score is the expected Z-score of the data value if the distribution of the random variable is normal. The expected Z-score of an observed value will depend upon the number of observations in the data set. 7-57 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Drawing a Normal Probability Plot Step 1 Arrange the data in ascending order. i 0.375 Step 2 Compute fi n 0.25 where i is the index (the position of the data value in the ordered list) and n is the number of observations. The expected proportion of observations less than or equal to the ith data value is f. 5-58 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. Drawing a Normal Probability Plot Step 3 Find the z-score corresponding to fi from Table V. Step 4 Plot the observed values on the horizontal axis and the corresponding expected z-scores on the vertical axis. 5-59 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. The idea behind finding the expected z-score is that, if the data comes from normally distributed population, we could predict the area to the left of each of the data value. The value of fi represents the expected area left of the ith observation when the data come from a population that is normally distributed. For example, f1 is the expected area to the left of the smallest data value, f2 is the expected area to the left of the second smallest data value, and so on. 7-60 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. If sample data is taken from a population that is normally distributed, a normal probability plot of the actual values versus the expected Zscores will be approximately linear. 7-61 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. We will be content in reading normal probability plots constructed using the statistical software package, MINITAB. In MINITAB, if the points plotted lie within the bounds provided in the graph, then we have reason to believe that the sample data comes from a population that is normally distributed. 7-62 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Interpreting a Normal Probability Plot The following data represent the time between eruptions (in seconds) for a random sample of 15 eruptions at the Old Faithful Geyser in California. Is there reason to believe the time between eruptions is normally distributed? 728 730 726 7-63 678 722 716 723 708 736 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 735 708 736 735 714 719 The random variable “time between eruptions” is likely not normal. 7-64 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. 7-65 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Interpreting a Normal Probability Plot Suppose that seventeen randomly selected workers at a detergent factory were tested for exposure to a Bacillus subtillis enzyme by measuring the ratio of forced expiratory volume (FEV) to vital capacity (VC). NOTE: FEV is the maximum volume of air a person can exhale in one second; VC is the maximum volume of air that a person can exhale after taking a deep breath. Is it reasonable to conclude that the FEV to VC (FEV/VC) ratio is normally distributed? Source: Shore, N.S.; Greene R.; and Kazemi, H. “Lung Dysfunction in Workers Exposed to Bacillus subtillis Enzyme,” Environmental Research, 4 (1971), pp. 512 - 519. 7-66 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE 7-67 Interpreting a Normal Probability Plot Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc. It is reasonable to believe that FEV/VC is normally distributed. (the probability plot is roughly linear) 7-68 Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.