STATS DOESN T SUCK! ~ CHAPTER 4

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1 CHAPTER 4 QUESTION 1 The Geometric Mean Suppose you make a 2-year investment of $5,000 and it grows by 100% to $10,000 during the first year. During the second year, however, the investment suffers a 50% loss, from $10,000 back to $5,000. a) Calculate the arithmetic mean. b) Calculate the geometric mean. c) Compare the values of the arithmetic and geometric means. 1

2 CHAPTER 4 QUESTION 2 More Geometric Mean The total sales of a company over a six year period are shown in the accompanying table. Year Sales ($millions) a) Calculate the five annual growth rates from year 1 to year 6. b) Find the geometric mean growth rate in sales over this period. c) Find the arithmetic mean growth rate in sales over this period. d) What is the best estimate of the growth rate in sales in year 7? 2

3 CHAPTER 4 QUESTION 3 Empirical Rule A supermarket has determined that daily demand for egg cartons has an approximate bell shaped distribution, with a mean of 55 cartons and a standard deviation of six cartons. a) How often can we expect between 49 and 61 cartons to be sold in a day? (Give a percentage.) b) What percentage of the time will the number of cartons of eggs sold be more than 2 standard deviations from the mean? c) If the supermarket begins each morning with a supply of 67 cartons of eggs, how often will demand exceed the supply? (Give a percentage.) 3

4 CHAPTER 4 QUESTION 4 Chebyshev's Theorem & Empirical Rule A professor has announced that the grades on a statistics exam have a mean value of 72 and a standard deviation of 6. Not knowing anything about the shape of the distribution of grades, what can we say about the proportion of grades that are between: a) 66 and 78? b) 60 and 84? c) 54 and 90? d) What would the answers to a), b), and c) be if the professor also announced that the grades have a mound-shaped distribution? 4

5 CHAPTER 4 QUESTION 5 Measures of Central Location & Variability The following data represent the weight loss in a sample of 10 women from a certain health club Please show all of your calculations in the space provided below and place your answers in the appropriate boxes. a. Mean b. Median c. Mode d. Variance e. Standard deviation f. Coefficient of Variation g) M.A.D. I added this because it is often asked on exams. 5

6 CHAPTER 4 Multiple Choice STATS DOESN T SUCK! ~ CHAPTER 4 1. The following are the grades a professor gave on the first test in a statistics class: 52, 90, 88, 61, 75, 82, 75, 83, 88, and 86. What was the median score on this test? a) 82 b) 82.5 c) 78.5 d) 88 e) A clothes store manager has sales data of trouser sizes for the last month's sales. Which measure of central tendency should the manager use, if the manager is interested in the most sellable size? a) Mean b) Median c) Mode d) Standard deviation e) Variance 3. What measure of central tendency is most sensitive to skewness? a) Mode b) Median c) Mean d) They are all about the same e) Median and Mean 4. The number of bottles filled each day in a bottling plant is observed for a week: 674, 589, 613, 689, 554, 586, and 594. What was the mean? a) 585 b) 670 c) d) e) None of the above 6

7 5. The number of bottles filled each day in a bottling plant is observed for a week: 674, 589, 613, 689, 554, 586, and 594. What was the standard deviation? a) 24 b) c) d) e) If a given distribution is known to be bell-shaped and symmetric, what percent of the observations are expected to fall within two standard deviations of the mean? a) 68% b) 75% c) 89% d) 95% e) Almost all 7. The only legitimate measure of central tendency for nominal scale data distributions is the: a) Mean b) Median c) Mode 8. The mean is an inappropriate measure of central tendency or most typical value for ordinal data. a) always b) often c) sometimes d) rarely e) never 7

8 9. Since the population size is always larger than the sample size, then the sample statistic a) can never be larger than the population parameter b) can never be equal to the population parameter c) can be smaller, larger, or equal to the population parameter d) can never be smaller than the population parameter 10. The hourly wages of a sample of 130 system analysts are given below. mean = 60 range = 20 mode = 73 variance = 324 median = 74 The coefficient of variation equals a) 0.30% b) 30% c) 5.4% d) 54% 11. The variance of a sample of 169 observations equals 576. The standard deviation of the sample equals a) 13 b) 24 c) 576 d) 28, The difference between the largest and the smallest data values is the a) variance b) standard deviation c) range d) coefficient of variation 13. If a data set has an even number of observations, the median a) cannot be determined b) is the average value of the two middle items c) must be equal to the mean d) is the average value of the two middle items when all items are arranged in ascending order 8

9 14. When data are positively skewed, the mean will usually be a) greater than the median b) smaller than the median c) equal to the median d) positive 15. If the variance of a data set is correctly computed with the formula using n - 1 in the denominator, which of the following is true? a) The data set is a sample b) The data set is a population c) The data set could be either a sample or a population d) The data set is from a census 16. The measure of dispersion that is influenced most by extreme values is a) the variance b) the standard deviation c) the range d) the median 17. The numerical value of the standard deviation can never be a) larger than the variance b) zero c) negative d) smaller than the variance 18. If two groups of numbers have the same mean, then a) their standard deviations must also be equal b) their medians must also be equal c) their modes must also be equal d) None of these alternatives is correct 9

10 19. The sum of deviations of the individual data elements from their mean is a) always greater than zero b) always less than zero c) sometimes greater than and sometimes less than zero, depending on the data elements d) always equal to zero 20. A numerical measure of linear association between two variables is the a) variance b) covariance c) standard deviation d) coefficient of variation 21. Positive values of covariance indicate a) a positive variance of the x values b) a positive variance of the y values c) the standard deviation is positive d) positive relation between the independent and the dependent variables 22. The coefficient of correlation ranges between a) 0 and 1 b) -1 and +1 c) minus infinity and plus infinity d) 1 and During a cold winter, the temperature stayed below zero for ten days (ranging from -20 to -5). The variance of the temperatures of the ten day period a) is negative since all the numbers are negative b) must be at least zero c) cannot be computed since all the numbers are negative d) can be either negative or positive 10

11 24. Since the mode is the most frequently occurring data value, it a) can never be larger than the mean b) is always larger than the median c) is always larger than the mean d) None of these alternatives is correct 25. This set of data represents the average rainfall (measured as inches) in June at the Statsville airport for the past 10 summers: 0.7, 1.6, 4.1, 0.7, 0.7, 3.3, 2.4, 1.6, 0.1 If the largest and smallest observed values were removed, in what way would the mean, median and mode be affected? a) The mean, median, and mode are unaffected. b) Both the mean and median will change, but the mode will not. c) Only the mean will change. d) The median is unaffected, but the mean and the mode will change. 26. The If a data set is approximately bell-shaped with µ = 40 and σ = 8, the values between 24 and 48 will consisit of what percent of the data? a) 68% b) 86% c) 81.5% d) 99.7% 27. The length of baseball games is approximately normally distributed with a mean of 2.45 hours. Using the Empirical Rule, about 95% of baseball games take between 1.71 and 3.19 hours to play. The standard deviation is a) 0.37 b) 10.4 c) 4.5 d) None of the above 11