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Statistics for-management-by-levin-and-rubin-solution-manualphpapp02 Download Now Download Download to read offline. Mahvesh Zahra Follow. Statistics for management. Applied Statistics And Doe Mayank. Basic concepts of cost accounting. Bad newsletters. Effects of unemployment on economy. What to Upload to SlideShare. Walter Azko had developed the computer company from a strange background. Unlike Lee, Walter never finished college. Walter had traveled extensively in the Far East with his parents, so it was only natural that he would begin an importing business while still a student at Boulder.
He imported just about anything that could be sold cheaply and that would appeal to students: furniture, gifts, household utensils, and some clothing.
On one buying trip to Taiwan in the early s, Walter was offered some personal computers. Looking back, they were awful. The computer business grew, and within 2 years, Walter sold his retail importing business and concentrated solely on importing and selling computers.
From this location, he could market directly to students at the Universities at Boulder, Fort Collins, and Greeley. The name Loveland Computers seemed a natural. At first, Walter Azko acted as his own sales staff, personally delivering computers from the back of his car. Walter made every sale on price alone and word-of-mouth referrals supplemented a few ads placed in the college newspapers. Because he sold directly to students and enthusiasts, it seemed that he was the only game in town.
At the top end of the market for PCs, IBM was using expensive retail distribution, targeting the business market. One or two other companies had found cheap overseas suppliers and they were pursuing a mail-order strategy.
Walter thought customers would be reluctant to buy such an expensive—and novel—piece of equip- ment sight unseen, but the arrival of a new shipment of computers with preinstalled hard disk drives gave him the motivation to run a few ads of his own. So Loveland Computers joined the ranks of the national mail-order box shops, and by , the company was one of the two dozen companies in this market.
But the market for PCs was huge and growing rapidly. Uncle Walter had become a rich man. Along the way, Walter Azko realized that to give customers exactly what they wanted, there were advantages in assembling computers at his ever-expanding Loveland facility.
He never saw himself as a manufacturer—just an assembler of premade parts such as drive controllers and power supplies. To configure new machines and to help with specifications, Walter hired a bright young engineer, Gratia Delaguardia. Gratia knew hardware: She had completed several development projects for Storage Technology.
In only a few years at Loveland Computers, she built a development staff of more than two dozen and was rewarded with a partnership in the business. Loveland Computers had a few setbacks due to misjudging demand. Walter Azko was always opti- mistic about sales so inventory of components was often much greater than needed. Gratia Delaguardia had concluded that Loveland ought to be able to manage the supplies better, but it seemed difficult to predict what the market would be like from one month to the next.
You know that things move pretty fast around here. Seems like each model lasts about 6 months and then we replace it with something fancier. Thing is, they want to know a whole lot about our sales growth: how much is coming from which products and so on. They want to know how long each model lasts, what we should project for next year.
Now, of course, I have monthly sales reports going back almost to the beginning. And, of course, no one wants to flip through, say, 48 months of reports. Your job is to organize it all so it makes sense when these city slickers come to town in their corporate jet.
These folks are due in next Monday. Study Questions: What information should Lee gather, other than financial information relating to sales and income?
Cumulative Frequency Distribution A tabular display of data showing how many observations lie above, or below, certain values. Data A collection of any number of related observations on one or more variables. Data Array The arrangement of raw data by observations in either ascending or descending order. Data Point A single observation from a data set. Data Set A collection of data.
Discrete Classes Data that do not progress from one class to the next without a break; that is, where classes represent distinct categories or counts and may be represented by whole numbers. Frequency Curve A frequency polygon smoothed by adding classes and data points to a data set.
Frequency Distribution An organized display of data that shows the number of observations from the data set that falls into each of a set of mutually exclusive and collectively exhaustive classes. Frequency Polygon A line graph connecting the midpoints of each class in a data set, plotted at a height corresponding to the frequency of the class. Histogram A graph of a data set, composed of a series of rectangles, each proportional in width to the range of values in a class and proportional in height to the number of items falling in the class, or the fraction of items in the class.
Ogive A graph of a cumulative frequency distribution. Open-Ended Class A class that allows either the upper or lower end of a quantitative classification scheme to be limitless. Population A collection of all the elements we are studying and about which we are trying to draw conclusions.
Raw Data Information before it is arranged or analyzed by statistical methods. Relative Frequency Distribution The display of a data set that shows the fraction or percentage of the total data set that falls into each of a set of mutually exclusive and collectively exhaustive classes. Representative Sample A sample that contains the relevant characteristics of the population in the same proportions as they are included in that population. Sample A collection of some, but not all, of the elements of the population under study, used to describe the population.
This formula uses the next value of the same units because it measures the inter- val between the first value of one class and the first value of the next class. Review and Application Exercises The following set of raw data gives income and education level for a sample of individuals. Would rearranging the data help us to draw some conclusions? Rearrange the data in a way that makes them more meaningful.
From its data, it constructed the following frequency distribution:. The company keeps records on the number of each item produced per month in order to examine the relative production.
Records show the following numbers of each item were produced by the company for the last month of 20 operating days: 9, 10, 10, 9, 9, 10, 10, 9, 9, 9, 10, 10, 9, 9, 10, Construct an ogive that will help you answer these questions. The following data were collected during a typical day: Waiting Time Minutes 12 16 21 20 24 3 11 17 29 18 26 4 7 14 25 1 27 15 16 5.
What comment can you make about patient waiting time from your data array? What additional interpretation can you give to the data from the frequency distribution? If not, which sample is more representative, and why?
If we are to select a sample of individuals from this population, how many should be women to make our sample considered strictly representative?
Department of Labor publishes several classifications of the unemployment rate, as well as the rate itself. Recently, the unemployment rate was 6. Along with some other variables, he recorded the average number of miles run per day. He compiled his results into the following distribution: Miles per Day Frequency 1.
Comment on the adequacy of this survey. Age Group Relative Proportion in Population 12—17 0. A total of responses are received. Comment on the data available in these responses in terms of the five tests for data. For each of the numbered blanks on the card, determine the most likely characteristics of the categories that would be used by the company to record the informa- tion. In particular, would they be 1 quantitative or qualitative, 2 continuous or discrete, 3 open-ended or closed?
Briefly state the reasoning behind your answers. Address Where was appliance purchased? Zip Code Why was appliance purchased? Determine the class marks midpoints for each of the intervals. Suggest a better way to display these data. Explain why it is better. Why or why not? If not, what would the raw data be in this situation?
He asked Peter Wilson, a Ph. Peter compiled the following:. Specialty Faculty Members Publishing Accounting and statistics 6 Accounting and finance 3 Marketing and finance 8 Statistics and finance 9 Statistics and marketing 21 No publications 1 Construct a relative frequency distribution for the types of specialties.
Hint: The catego- ries of your distribution will be mutually exclusive, but any individual may fall into several categories.
Niles went from office to office and interviewed execu- tives eligible to participate. To examine the situation, Niles decides to construct both frequency and relative frequency distributions. Following are a frequency distribution and a relative frequency distribution of the number of interviews required per salesperson per sale. Fill in the missing data. Cline, the mine superintendent of the Grover Coal Co. How many were under the limit?
The values are in tons of coal mined per shift:. How many did better than expected? A recent delivery of bolts from a new supplier caught the eye of a clerk. Suboleski sent 25 of the bolts to a testing lab to determine the force necessary to break each of the bolts. In thousands of pounds of force, the results are as follows: What proportion withstood at least , pounds? What should Suboleski recommend the company do about continuing to order from the new supplier? This number, called the phone overflow rate, is expressed as a percentage of the total number of calls taken in a given week.
Loy has used the overflow data for the last year to prepare the following frequency distribution: Overflow Rate Frequency Overflow Rate Frequency 0. Karl Slayden has just been hired to help rebuild the company. He has found sales records for the last 2 months: Country of Sales Country of Sales Country of Sales 1 3 7 4 13 1 2 1 8 9 14 1 3 1 9 5 15 5 4 8 10 1 16 6 5 3 11 3 17 6 6 5 12 7 18 2 19 2 23 1 27 1 20 1 24 7 28 5 21 1 25 3 22 2 26 1 a Arrange the sales data in an array from highest to lowest.
Compare the two. Information was collected on the number of minutes that 3, consecutive drivers waited in line at the toll gates: Minutes of Waiting Frequency Minutes of Waiting Frequency less than 1 75 9— What percentage of the drivers had to wait more than 4 minutes in line? For example, today Maribor made 7, tiles and had a breakage rate during production of 2 percent.
To measure daily tile output and breakage rate, Olsen has set up equally spaced classes for each. The class marks midpoints of the class intervals for daily tile output are 4,, 5,, 6,, 6,, 7,, and 7, The class marks for breakage rates are 0. It has gathered the following information on the number of models of engines in different size categories used in the racing market it serves: Class Frequency Engine Size in Cubic Inches of Models — 1 — 7 — 7 — 8 — 17 — 16 — 15 — 7.
Construct a cumulative relative frequency distribution that will help you answer these questions: a Seventy percent of the engine models available are larger than about what size? They arrange a fact- finding trip to Denver, and in a meeting they are given the following frequency distribution of number of passengers per car: Number of Passengers Frequency 1—10 20 11—20 18 21—30 11 31—40 8 41—50 3 51—60 1.
What percentage of the total observations are more than 30 and less than 41 passengers? What proportion of trips would be economical and satisfying? Moreno wants an initial comparison to see whether waiting times at the toll plaza appear to have dropped.
Here are the waiting times observed for 3, consecutive drivers after the mill schedule change:. Minutes of Waiting Frequency less than 1 1— 2. Is there an obvious difference in waiting times? Construct a pie chart showing the distribution of type of bank account held by the people in the banks. Question 2 2. Construct a bar chart showing the frequency of usage of e-banking by the customers. Question 5 3. Construct a bar chart comparing the level of satisfaction with e-services across the different age group of customers.
Question 9 vs Question 14 4. Draw an appropriate chart depicting the problems faced in e-banking and the promptness with which they are solved. Question 10 vs Question 12 5. Draw an appropriate diagram to study the gap in the expected and observed e-banking services provided by the banks to their customers. Organize raw data into an array. Should data No be condensed and simplified? Prepare frequency distribution by grouping arrayed data into classes.
Do you want No a graphic display? Prapare graphic presentation of frequency distribution:. T he vice president of marketing of a fast-food chain is studying the sales performance of the stores in his eastern district and has compiled this frequency distribution of annual sales:. Sales s Frequency Sales s Frequency — 4 1,—1, 13 — 7 1,—1, 10 — 8 1,—1, 9 1,—1, 10 1,—1, 7 1,—1, 12 1,—1, 2 1,—1, 17 1,—1, 1. The vice president would like to compare the eastern district with the other three districts in the country.
To do so, he will summarize the distribution, with an eye toward getting information about the central tendency of the data. This chapter also discusses how he can measure the variability in a distribu- tion and thus get a much better feel for the data. In most cases, however, we need more exact measures. In these cases, Summary statistics, central we can use single numbers called summary statistics to describe tendency, and dispersion characteristics of a data set.
Two of these characteristics are particularly important to decision makers: central tendency and dispersion. Central Tendency Central tendency is the middle point of a Middle of a data set distribution. Measures of central tendency are also called measures of location. In Figure , the central location of curve B lies to the right of those of curve A and curve C. Notice that the central location of curve A is equal to that of curve C.
Dispersion Dispersion is the spread of the data in a Spread of a data set distribution, that is, the extent to which the observations are scattered.
Notice that curve A in Figure has a wider spread, or dispersion, than curve B. There are two other characteristics of data sets that provide useful information: skewness and kurto- sis. Although the derivation of specific statistics to measure these characteristics is beyond the scope of this book, a general understanding of what each means will be helpful. Skewness Curves representing the data points in the data Symmetry of a data set set may be either symmetrical or skewed.
Symmetrical curves, like the one in Figure , are such that a vertical line drawn from the center of the curve to the horizontal axis divides the area of the curve into two equal parts.
Each part is the mirror image of the other. Curves A and B in Figure are skewed curves. They are Skewness of a data set skewed because values in their frequency distributions are con- centrated at either the low end or the high end of the measuring scale on the horizontal axis.
The values are not equally distributed. Curve A is skewed to the right or positively skewed because it tails off toward the high end of the scale.
Curve B is just the opposite. It is skewed to the left negatively skewed because it tails off toward the low end of the scale. The curve would be skewed to the right, with many values at the low end and few at the high, because the inventory must turn over rapidly.
Similarly, curve B could represent the frequency of the number of days a real-estate broker requires to sell a house. It would be skewed to the left, with many values at the high end and few at the low, because the inventory of houses turns over very slowly. Curve A: Curve B: skewed right skewed left. Kurtosis When we measure the kurtosis of a distribution, we Peakedness of a data set are measuring its peakedness.
In Figure , for example, curves A and B differ only in that one is more peaked than the other. They have the same central location and dispersion, and both are symmetrical. Statisticians say that the two curves have different degrees of kurtosis. For the next two distributions, indicate which distribution, if any e Has values more evenly distributed across the range of possible values. This is true in cases such as the average winter temperature in New York City, the average life of a flashlight battery, and the average corn yield from an acre of land.
Table presents data describing the number of days the gen- The arithmetic mean erators at a power station on Lake Ico are out of service owing to regular maintenance or some malfunction.
To find the arithmetic mean, we sum the values and divide by the number of observations:. With this figure, the power plant manager has a reasonable single measure of the behavior of all her generators. Conventional Symbols To write equations for these measures of frequency distribu- Characteristics of a samp le are tions, we need to learn the mathematical notations used by stat- called statistics isticians. A sample of a population consists of n observations a lowercase n with a mean of x read x-bar.
Remember that the measures we compute for a sample are called statistics. The notation is different when we are computing measures Characteristics of a population for the entire population, that is, for the group containing every are called parameters element we are describing.
The number of. Generally in statistics, we use italicized Roman letters to symbolize sample information and Greek letters to symbolize population information. Calculating the Mean from Ungrouped Data In the example, the average of 8. It would sample means be x the sample mean if the 10 generators are a sample drawn from a larger population of generators. To write the formulas for these two means, we combine our mathematical symbols and the steps we used to determine the arithmetic mean.
If we add the values of the observations and divide this sum by the number of observations, we will get. Population Arithmetic Mean Sum of values of all observations. Sample Arithmetic Mean Sum of values of all observations. Similarly, x is the sample arithmetic mean and n is the number of observations in the sample. Notice that to calculate this mean, we added every observation. Dealing with ungrouped data Statisticians call this kind of data ungrouped data.
The computa- tions were not difficult because our sample size was small. But suppose we are dealing with the weights of 5, head of cattle and prefer not to add each of our data points separately.
Or suppose we have access to only the frequency distribution of the data, not to every individual observation. In these cases, we will need a different way to calculate the arithmetic mean. Calculating the Mean from Grouped Data A frequency distribution consists of data that are grouped by Dealing with grouped data classes. Each value of an observation falls somewhere in one of the classes. Unlike the SAT example, we do not know the sepa- rate values of every observation.
Suppose we have a frequency distribution illustrated in Table of average monthly checking-account balances of customers at a branch bank. From the information in this table, we can easily compute an estimate of the value of the mean of these grouped data.
It is an estimate because we do not use all data points in the sample. Had we used the original, ungrouped data, we could have calculated the actual value of the mean, but Estimating the mean only by averaging the separate values. For ease of calcula- tion, we must give up accuracy. To find the arithmetic mean of grouped data, we first calculate Calculating the mean the midpoint of each class. To make midpoints come out in whole cents, we round up.
Thus, for example, the midpoint for the first class becomes Then we multiply each midpoint by the frequency of observa- tions in that class, sum all these results, and divide the sum by the total number of observations in the sample. The formula looks like this:. Class Dollars Frequency 0— This is our approximation from the frequency distribution.
Notice that because we did not know every data point in the sample, we assumed that every value in a class was equal to its midpoint. Our results, then, can only approximate the actual average monthly balance. Coding In situations where a computer is not available and we have to do Assigning codes to the the arithmetic by hand, we can further simplify our calculation of midpoints the mean from grouped data.
Using a technique called coding, we eliminate the problem of large or inconvenient midpoints. Instead of using the actual midpoints to perform our calculations, we can assign small-value consecutive inte- gers whole numbers called codes to each of the midpoints. The integer zero can be assigned anywhere, but to keep the integers small, we will assign zero to the midpoint in the middle or the one nearest to the middle of the frequency distribution.
Then we can assign negative integers to values smaller than that midpoint and positive integers to those larger, as follows:. Symbolically, statisticians use x0 to represent the midpoint Calculating the mean from that is assigned the code 0, and u for the coded midpoint. Table illustrates how to code the midpoints and find the sample mean of the annual snowfall in inches over 20 years in Harlan, Kentucky. Advantages and Disadvantage s of the Arithmetic Mean The arithmetic mean, as a single number representing a whole Advantages of the mean data set, has important advantages.
First, its concept is familiar to most people and intuitively clear. Second, every data set has a mean. It is a measure that can be calculated, and it is unique because every data set has one and only one mean. Finally, the mean is useful for performing statistical procedures such as comparing the means from several data sets a procedure we will carry out in Chapter 9.
Yet, like any statistical measure, the arithmetic mean has dis- Three disadvantages of the advantages of which we must be aware. First, although the mean is reliable in that it reflects all the values in the data set, it may mean also be affected by extreme values that are not representative of the rest of the data.
Notice that if the seven members of a track team have times in a mile race shown in Table , the mean time is. A second problem with the mean is the same one we encountered with our checking-account balances: It is tedious to compute the mean because we do use every data point in our calculation unless, of course, we take the short-cut method of using grouped data to approximate the mean.
The mean or average can be an excellent measure of central tendency how data group around the middle point of a distribution. But unless the mean is truly representative of the data from which it was computed, we are violating an important assumption.
A helpful hint in choosing which one of these to compute is to look at the data points. Class Frequency Class Frequency The loan showed the following ending monthly balances last year:. Does it qualify? Applications Child-Care Community Nursery is eligible for a county social services grant as long as the average age of its children stays below 9. If these data represent the ages of all the children currently attending Child-Care, do they qualify for the grant?
The family incomes of the attending children are. Time in seconds Frequency 20—29 6 30—39 16 40—49 21 50—59 29 60—69 25 70—79 22 80—89 11 90—99 7 — 4 — 0 — 2. The owner will build if the aver- age number of animals sold during the first 6 months of is at least and the overall monthly average for the year is at least The data for are as follows:. May June July Aug. The resulting volumes in ounces for the trials were as follows:.
The company does not normally recalibrate the filling machine for this cologne if the average volume is within 0. Should it recalibrate? Using a stopwatch and observing the platemakers, he collects the following times in seconds.
An average per-plate time of less than Should the production manager be concerned? Smith travels the eastern United States as a sales representative for a textbook publisher. She is paid on a commission basis related to volume. Her quarterly earnings over the last 3 years are given below.
Furthermore, show that both these numbers equal the mean of all 12 numbers in the data table. This is M. She con- tends that during her tenure she has managed the book-mobile repair budget better than her predecessor did.
Here are data for bookmobile repair for 15 years:. Has she been saving the county money? Consider, for example, the company in Table , which uses. The company wants to know the average cost of labor per hour for each of the products. A simple arithmetic average of the labor wage rates would be. But these answers are incorrect. To be correct, the answers must take into account that differ- ent amounts of each grade of labor are used.
We can determine the correct answers in the following manner. Another way to calculate the correct average cost per hour The correct answer is the for the two products is to take a weighted average of the weighted mean cost of the three grades of labor. To do this, we weight the hourly wage for each grade by its proportion of the total labor required to produce the product.
One unit of product 1, for example, requires 8 hours of labor. Unskilled labor uses 18 of this time, semiskilled labor uses 2 8 of this time, and skilled labor requires 5 8 of this time.
If we use these fractions as our weights, then one hour of labor for product 1 costs an average of. Similarly, a unit of product 2 requires 10 labor hours, of which 4 10 is used for unskilled labor, for semiskilled labor, and for skilled labor. By using these fractions as weights, one hour of labor for product 2 costs. Thus, we see that the weighted averages give the correct values for the average hourly labor costs of the two products because they take into account that different amounts of each grade of labor are used in the products.
Symbolically, the formula for calculating the weighted average is. When we calculated the arithmetic mean data: the weighted mean from grouped data page 79 , we actually found a weighted mean, using the midpoints for the x values and the frequencies of each class as the weights. We divided this answer by the sum of all the frequencies, which is the same as dividing by the sum of all the weights. In like manner, any mean computed from all the values in a data set according to Equation or is really a weighted average of the components of the data set.
What those components are, of course, determines what the mean measures. In a factory, for example, we could determine the weighted mean. The lowercase w is called a subscript and is a reminder that this is not an ordinary mean but one that is weighted according to the relative importance of the values of x. Distinguish between distinct values and individual observations in a data set, since several obser- vations can have the same value.
If values occur with different frequencies, the arithmetic mean of the values as opposed to the arithmetic mean of the observations may not be an accurate measure of central tendency.
In such cases, we need to use the weighted mean of the values. If you are using an average value to make a decision, ask how it was calculated. If the values in the sample do not appear with the same frequency, insist on a weighted mean as the correct basis for your decision. The items cost. Is Dave getting himself into or out of trouble by talking about weighted averages? SC Bennett Distribution Company, a subsidiary of a major appliance manufacturer, is forecasting regional sales for next year.
What is the average rate of sales growth forecasted for next year? Applications A professor has decided to use a weighted average in figuring final grades for his seminar students. From the following data, compute the final average for the five students in the seminar.
Student Homework Quizzes Paper Midterm Final 1 85 89 94 87 90 2 78 84 88 91 92 3 94 88 93 86 89 4 82 79 88 84 93 5 95 90 92 82 Each case contains 24 tapes. Is this a good business practice for Jim? The fol- lowing frequency distribution resulted;.
What is the average number of times a subscriber saw a Keyes advertisement during December? The production forecast for the next year has been completed. The Orlando division, with yearly production of 72 million windows, is predicting an The Pittsburgh division, with yearly production of 62 million, should grow by 6. The Seattle division, with yearly production of 48 million, should also grow by 6.
The Minneapolis and Dallas divisions, with yearly productions of 89 and 94 million windows, respectively, are expecting to decrease production in the coming year by 9. What is the average rate of change in production for the Nelson Window Company for the next year? Postal Service handles seven basic types of letters and cards: third class, second class, first class, air mail, special delivery, registered, and certified.
The mail volume during is given in the following table:. Office records indicate the following number of hours billed last year in each category: 8,, 14,, 24,, and 35, If Matthews, Young is trying to come up with an average billing rate for estimating client charges for next year, what would you suggest they do and what do you think is an appropriate rate? What we need to find is the geometric mean, simply called the G.
Consider, for example, the growth of a savings account. The growth is summarized in Table The growth factor is the amount by which we multiply the sav- In this case, the arithmetic ings at the beginning of the year to get the savings at the end mean growth rate is incorrect of the year. The simple arithmetic mean growth factor would be 1. Thus, the correct average growth factor must be slightly less than 1. The result is the geometric mean growth rate, which is the appropriate average to use here.
The formula for finding the geometric mean of a series of numbers is. Geometric Mean Number of x values [] G. If we apply this equation to our savings-account problem, we can determine that 1. Notice that the correct average interest rate of This happens because the interest rates are relatively small. Be careful however, not to be tempted to use the arithmetic mean instead of the more compli- cated geometric mean. The following example demonstrates why. In highly inflationary economies, banks must pay high interest rates to attract savings.
Suppose that over 5 years in an unbelievably inflationary economy, banks pay interest at annual rates of , , , , and percent, which correspond to growth factors of 2, 3, 3. This corresponds to an average interest rate of percent. In this case, the use of the appropriate mean does make a significant difference.
We use the geometric mean to show multiplicative effects over time in compound interest and infla- tion calculations. In certain situations, answers using the arithmetic mean and the geometric mean will not be too far apart, but even a small difference can generate a poor decision. A good working hint is to use the geometric mean whenever you are calculating the average percentage change in some variable over time.
Calculate the average percentage increase in bad-debt expense over this time period. If this rate continues, estimate the percentage increase in bad debts for , relative to SC Realistic Stereo Shops marks up its merchandise 35 percent above the cost of its latest addi- tions to stock. Applications Hayes Textiles has shown the following percentage increase in net worth over the last 5 years:.
What is the average percentage increase in net worth over the 5-year period? Calculate the average percentage change in net worth over this time period. Cummings, William H. Mason, Scott A. Morton, David R. Rambure, A. Incredibly Easy! FRCP Lond. FRCP Edin. McArdle BS M. Ed PhD, Frank I. Katch, Victor L. Fowler MD. Levin, David S. Basavaraju, C. NET Core 1. One-Volume By David E. Shi, George Brown Tindall. By Kike Calvo. Human Interest. Stage 1 By Jennifer Bassett.
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