The Normal Distribution

Transcription

1 CHAPTER 6 The Normal Distribution CHAPTER OUTLINE 6.1 The Standard Normal Distribution 6.2 Probability Calculations with the Normal Distribution 6.3 Applications of the Normal Distribution 6.4 Determining the z and x values when an Area under the Normal Distribution Curve is Known 6.5 Transforming Observations to Attain Near Normality 6.1 THE STANDARD NORMAL DISTRIBUTION The standard normal distribution is a normal distribution that has a mean equal to 0 and a standard deviation of 1. Probabilities and z-scores associated with this distribution can be found with Excel s NORMSDIST and NORMSINV functions, respectively. The S in the middle of each function name is for Standard. First, we ll consider computing probabilities for the standard normal distribution. (In Section 6.4, we ll calculate z-scores.) Your textbook has a table of probabilities associated with various z-scores that can be used, but Excel can calculate probabilities for any z-score with its built-in NORMSDIST function. This function calculates the cumulative probability for any z-score, i.e., the area under the curve to the left of that z- score. So, if you want the probability between two z-scores, you have to use the function for each one and then subtract, greater minus smaller. (Your answer, which is a probability or area, should always be non-negative.) 101

2 102 Chapter 6 The Normal Distribution Example 6-1 Find the area under the standard normal curve between z = 0 and z = Solution: Click on a blank cell in an Excel worksheet. Type the following: =NORMSDIST(1.95)-NORMSDIST(0). Hit ENTER, and you should see the probability of appear. The other option is to insert the NORMSDIST function, type 1.95 into the text box, click OK, hit F2 to edit the cell, type a minus sign after the last parenthesis, insert NORMSDIST again, type 0, and then click OK again. Example 6-2 Use Excel to find the following areas under the standard normal curve. a) Area to the right of z = 2.32 b) Area to the left of z = 1.54 Solution: For part a), you need to subtract the cumulative probability (area to the left) from 1 to get the complementary probability (area to the right). Type the following into a blank cell: =1-NORMSDIST(2.32). You could, instead, just type =1- and then insert the function and enter Or you could find the cumulative probability in one cell, and then in another cell type =1- and then click on the cell with the cumulative probability, and hit ENTER. Either way you do it, you should see the probability of appear. For part b), just insert the NORMSDIST function (or type it in with = before it and ( after it) and enter This should give you the desired cumulative probability (area to the left) of PROBABILITY CALCULATIONS WITH THE NORMAL DISTRIBUTION If a random variable x is normal with mean µ and standard deviation σ, then the z- x µ score, defined by z =, is normal with mean 0 and standard deviation 1. This is σ important, especially when comparing populations which have different means and/or standard deviations. One can use Excel s STANDARDIZE function to find the z-score that corresponds to some x-value from a nonstandard normal distribution (where the mean is not 0 or the standard deviation is not 1). This function has three inputs: 1) the x-value to be standardized = X, 2) the mean of the normal distribution from which the x-value is taken = Mean, 3) the standard deviation of the normal distribution from which the x-value is taken = Standard_dev.

3 Excel Manual 103 Example 6-3 Let x be a continuous random variable that has a normal distribution with a mean of 50 and a standard deviation of 10. Convert the following x-values to z-scores. a) x = 55 b) x = 35 Solution: Click on a blank cell in an Excel worksheet, and insert the STANDARDIZE function. For part a), enter 55 into the first text box, 50 into the second, and 10 into the third. (You can hit TAB to go from one text box to the next, if you wish.) Click OK or hit ENTER, and you should see the z-score of 0.5 appear. Figure 6.1 Use Excel s STANDARDIZE function if you want to convert an x-value from a normal distribution with given mean and standard deviation to a z-score. For part b), you can simply edit the first input (double-click on the cell or click on it and hit F2). Change the 55 to a 35 and hit ENTER. You should see a z-score of 1.5 appear. Figure 6.2 Excel s STANDARDIZE function can be typed in and/or edited instead of inserted. In order to compute probabilities associated with a nonstandard normal distribution (when the mean is not 0 or the standard deviation is not 1), you can either first standardize and use the NORMSDIST function, or you can more simply use the NORMDIST function which takes into account the different mean and standard deviation. (Notice the absence of the S in the middle of the function name, since this is used for a normal distribution that is not standard.) Specifically, Excel s NORMDIST function requires four inputs:

4 104 Chapter 6 The Normal Distribution 1) the x-value at the edge of the desired cumulative probability = X, 2) the mean of the distribution = Mean, 3) the standard deviation of the distribution = Standard_dev, 4) 1 (for TRUE) since we always want the cumulative normal probability = Cumulative. Note: The last input above is different from the discrete distributions like the binomial distribution, where the FALSE option is sometimes desirable. Keep in mind that the FALSE option would always generate a probability of zero for a continuous random variable. Example 6-4 Let x be a continuous random variable that is normally distributed with a mean of 25 and a standard deviation of 4. Find the area a) between x = 25 and x = 32 b) between x = 18 and x = 34. Solution: a) Click on a blank cell in an Excel worksheet. Type the following: =NORMDIST(32,25,4,1)-NORMDIST(25,25,4,1). Hit ENTER and you should see the probability of appear. The other option is to insert the NORMDIST function, type 32 into the first text box, 25 into the second, 4 into the third, and 1 into the fourth text box, click OK, hit F2 to edit the cell, type a minus sign after the last parenthesis, insert NORMDIST again, enter 25, 25, 4, and 1, and then click OK again. (You can hit TAB to go from one text box to the next, if you wish.) b) Double-click on the cell from part a) and edit the functions so that the first inputs are 34 and 18 instead of 32 and 25, respectively. Hit ENTER and you should see the probability of appear. Figure 6.3 Using the difference of two applications of Excel s NORMDIST function to find a probability between two values. Example 6-5 Let x be a continuous random variable that has a normal distribution with a mean of 80 and a standard deviation of 12. Find the area under the normal distribution curve to the left of 27. Solution: This is a direct application of the NORMDIST function, since we want the area to the left. Insert this function into a cell and enter the four inputs as shown in the figure below. Click OK, and you should see a value of E-06 appear. This is Excel s scientific notation, which means or Pretty small!

5 Excel Manual 105 That s because 27 is more than 4 standard deviations (4 12 = 48) below the mean (80). So, the area to the left of that is nearly 0. Figure 6.4 Excel gives the result in scientific notation when the probability is extremely small. 6.3 APPLICATIONS OF THE NORMAL DISTRIBUTION Now, we consider a couple of applications involving the use of Excel s NORMDIST function. Example 6-6 According to Automotive Lease Guide, the Porsche 911 sports car is among the vehicles that hold their value best. A Porsche 911 (with a price of $87,500 for a new car) is expected to command a price of $48,125 after three years (The Wall Street Journal, August 6, 2002). Suppose the prices of all three-year-old Porsche 911 sports cars have a normal distribution with a mean price of $48,125 and a standard deviation of $1600. Find the probability that a randomly selected three-year-old Porsche 911 will sell for a price between $46,000 and $49,000. Solution: Note that we have a normal distribution with a mean of 48,125 and a standard deviation of So, in order to find the probability, we need to use the NORMDIST function. The values and 1600 will be the 2 nd and 3 rd inputs, respectively. Since we seek the probability for between these two values, we need to apply the function twice, once with the upper value and once with the lower value as the first input, and then subtract. This can be done simply by typing in the following: =NORMDIST(49000,48125,1600,1)-NORMDIST(46000,48125,1600,1).

6 106 Chapter 6 The Normal Distribution You should see the resulting probability of , or approximately 61.6%. Important Note: You must not type commas within a number. Commas must only be used to separate the four distinct inputs. Of course, an alternative option is to insert the NORMDIST function, type into the first text box, into the second, 1600 into the third, and 1 into the fourth text box, click OK, hit F2 to edit the cell, type a minus sign after the last parenthesis, insert NORMDIST again, enter 46000, 48125, 1600, and 1, and then click OK again. The advantage of inserting the function is that you are prompted as to what the input(s) represent, in case you don t remember what goes where. Figure 6.5 Subtract two applications of the NORMDIST function in order to find the probability between two values. Example 6-7 A racing car is one of the many toys manufactured by Mack Corporation. The assembly times for this toy can be modeled using a normal distribution with a mean of 55 minutes and a standard deviation of 4 minutes. The company closes at 5 P.M. every day. If one worker starts to assemble a racing car at 4 P.M., what is the probability that she will finish this job before the company closes for the day? Solution: Identify the mean and standard deviation: 55 and 4 minutes, respectively. These are the 2 nd and 3 rd inputs of Excel s NORMDIST function. The 1 st input is the cutoff for the probability desired. In this case, we want the probability that she will finish in 1 hour (60 minutes) or less. So, the first input is 60. Always enter 1 for the last input. Hit ENTER or click on OK and you will get the cumulative probability of 60 or less. This value, (approximately 89.4%), is exactly what we want.

7 Excel Manual 107 Figure 6.6 Using Excel s NORMDIST function to find a cumulative probability. 6.4 DETERMINING THE Z AND X VALUES WHEN AN AREA UNDER THE NORMAL DISTRIBUTION CURVE IS KNOWN In order to go from a cumulative probability back to a z-score, use the function NORMSINV (with an S ). In order to go from a cumulative probability back to an x- value in a nonstandard normal distribution, you need to use the function NORMINV (with no S ). (Note that the INV suffix is in reference to an inverse operation i.e., going backwards.) The key is to make sure that you first know the cumulative area or probability, i.e. the area of the region under the curve lying to the left of the desired value. Example 6-8 Find the value of z such that the area under the standard normal curve in the right tail is Solution: Since the area to the right of z is.0050, the area to the left of z, the cumulative area, must be = Insert the NORMSINV function and enter.9950 (or you can simply enter ). Click on OK and you should see the z-score of appear.

8 108 Chapter 6 The Normal Distribution Figure 6.7 Use Excel s NORMSINV function to go backwards from a cumulative area (to the left of a z-score in the standard normal distribution) to the z-score. The first input of the NORMINV function is also the cumulative probability (or area to the left) of the desired value, but there are two more inputs required: the mean and standard deviation of the nonstandard normal distribution. (These two inputs were unnecessary for NORMSINV since by definition of the standard normal, the mean and standard deviation are always 0 and 1, respectively. As such, it is really just a special case of NORMINV.) Example 6-9 It is known that the life of a calculator manufactured by Intal Corporation has a normal distribution with a mean of 54 months and a standard deviation of 8 months. What should the warranty period be to replace a malfunctioning calculator if the company does not want to replace more than 1% of all the calculators sold? Solution: Note that we have a normal distribution with a mean of 54 and standard deviation of 8. We want to find a cutoff value for the 1% proportion (or probability). To do so, we need to determine the cumulative probability the probability to the left of the cutoff that we re looking for. If the 1% are being replaced, it s because they are not lasting long enough. They are the ones on the low end of the distribution. So, this is our cumulative probability: Symbolically, we need to find the number of months x such that PX ( x) = 0.01, where X is a normal random variable with µ = 54 and σ = 8. To perform the necessary computations on Excel, insert the NORMINV function and enter the three inputs 0.01, 54, and 8. Then, hit ENTER and you should see the value of , standing for approximately 35.4 months. This is what the warranty period should be.

9 Excel Manual 109 Figure 6.8 Use Excel s NORMINV function to find a cutoff value for a known cumulative proportion or probability in a nonstandard normal distribution. Example 6-10 Nearly all high school students who intend to go to college take the SAT test. In 2002, the mean SAT score (in verbal and mathematics) of all students was Debbie is planning to take this test soon. Suppose the SAT scores of all students who take this test with Debbie will have a normal distribution with a mean of 1020 and a standard deviation of 153. What should her score be on this test so that only 10% of all examinees score higher than she does? Solution: In this case, we have a normal distribution with a mean of 1020 and standard deviation of 153. Since we want to find a cutoff value for the 10% proportion (or probability), we need to use the NORMINV function. But, this is not the cumulative probability. Rather, we seek the percentage above the value we re looking for. As such, the cumulative probability is 100%-10% = 90% or Symbolically, we need to find the score x such that PX ( x) = 0.90, where X is a normal random variable with µ = 1020 and σ = 153. (Notice that this is talking about the 90 th percentile!) To perform the necessary computations on Excel, insert the NORMINV function and enter the three inputs 0.90, 1020, and 153. Hit ENTER and you should see the value of So, she needs to earn a score of at least Figure 6.9 Using Excel s NORMINV function to find the 90 th percentile in a normal distribution.

10 110 Chapter 6 The Normal Distribution 6.5 TRANSFORMING OBSERVATIONS TO ATTAIN NEAR NORMALITY Many powerful tools of statistical inference require that the underlying population from which samples are drawn to be approximately normal. If the samples are chosen from a population that departs from normality, a clever trick that sometimes works is to transform the data to a new variable whose distribution IS normal. If a data set doesn t have a symmetric histogram, then there are some standard transformations to apply (see page 249 of the text). However, you must keep in mind that this is far from an exact science. In fact, a bit of trial and error, and some good luck, plays an important role in transforming some data sets! We illustrate the approach with an example from your text, as follows: Example 6-11 A forester records the volume of timber, measured in cords, for 49 plots selected in a large forest. The data are tabulated below. An inspection of the corresponding histogram shows a long tail to the right. In order to transform such a histogram into one that is more normal, transform the data set using each of the following 4 1 transforms: x, x, log x, and x. Comment on the results Solution: First, enter this data into column A, along with the label Volume of Timber (in cords) (in cell A1). Then, in column B, type the label SQRT(x) in cell B1. Then, in cell B2, type =SQRT(A2), then highlight cells B2 B50, and go to Edit> Fill down to compute the square root of the data in column A. Similarly, type the labels x^(1.4), log(x), and 1/x in cells C1 E1, respectively. Then, compute the transformations in the same manner. The following 5 columns of data is what you should arrive at:

11 Excel Manual 111 Figure 6.10 The various transforms applied to the data in Example Now, simply we need to simply create frequency histograms for each column of data. Since the data is not feasibly broken down into single-valued classes, we need to create classes for each column. As such, for each column (separately), do the following: 1. Sort the data numerically. 2. Identify the maximum and minimum values, and calculate the range. 3. Divide the data into 6 classes, say determine the upper limits for each class (bin) as illustrated in Section Now, use Tools> Data Analysis> Histogram for each. The following graphs are the result of the above instructions.

12 112 Chapter 6 The Normal Distribution Figure 6.11 Frequency Histograms for each of the transformed data sets in Example It appears that the LOG transformation was the most successful in transforming our original skewed data set to a nearly normal one. Exercises 6.1 Use Excel s NORMSDIST function to find the area under the standard normal curve a) between z = 0 and z = 1.85 b) between z = 1.15 and z = 2.37 c) from z = 2.88 to z = 1.53 d) from z = 1.67 to z = 2.44 e) to the right of z = 1.56 f) to the left of z = Use Excel s STANDARDIZE function in order to find the z-score for each of the following x-values for a normal distribution with µ = 30 and σ = 5. a) x = 39 b) x = 17 c) x = 22 d) x = 42

13 Excel Manual Use Excel s NORMDIST function to find the following areas under a normal distribution curve with µ = 20 and σ = 4. a) Area between x = 20 and x = 27. b) Area from x = 23 to x = 25. c) Area between x = 9.5 and x = Use Excel s NORMDIST function in order to determine the area under a normal distribution curve with µ = 55 and σ = 7 a) to the right of x = 58 b) to the left of x = The U.S. Bureau of Labor Statistics conducts periodic surveys to collect information on the labor market. According to one such survey, the average earnings of workers in retail trade were $10 per hour in August 2002 (Bureau of Labor Statistics News, September 18,2002). Assume that the hourly earnings of such workers in August 2002 had a normal distrbution with a mean of $10 and a standard deviation of $1.10. Use Excel s NORMDIST function to find the probability that the hourly earnings of a randomly selected retail trade worker in August 2002 were a) more than $12 b) between $8.50 and $ The transmission on a model of a specific car has a warranty for 40,000 miles. It is known that the life of such a transmission has a normal distribution with a mean of 72,000 miles and a standard deviation of 12,000 miles. Use Excel s NORMDIST function to find the following. a) What percentage of the transmissions will fail before the end of the warranty period? b) What percentage of the transmissions will be good for more than 100,000 miles? 6.7 Use Excel s NORMSINV function in order to determine the value of z so that the area under the standard normal curve a) in the right tail is.0500 b) in the left tail is.0250 c) in the left tail is.0100 d) in the right tail is Fast Auto Service provides oil and lube service for cars. It is known that the mean time taken for oil and lube service at this garage is 15 minutes per car and the standard devation is 2.4 minutes. The management wants to promote the business by guaranteeing a maximum waiting time for its customers. If a customer s car is not serviced within that period, the customer will receive a 50% discount on the charges. The company wants to limit this discount to at most 5% of the customers. Assume that the times taken for oil and lube service for all cars have a normal distribution. Use Excel s NORMINV function to find what the maximum guaranteed waiting time should be. 6.9 Apply the four different transformations mentioned in Section 6.7 to the data set in Example 2-5. Comment on the results.