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21-1 Questions (a) to (c) refer to the following figure:

40-2 Numerator

41-1 Denominator

46-4 which of the following two events is the more unusual

50-1 In what sense is the mean

715-1 Which are the correct real limits for the frequency table given below?

717-1 What can be said about these two classes?

723-1 A large mass of data can best be summarized pictorially by

1014-1 "For a symmetric distribution, the mean and median are"

1016-1 The mean and median for this data are:

1021-1 "increases 3 points, the median will become"

1403-1 "Find S**2, the sample variance."

1562-1 If you are told a population has a mean of 25 and a variance of 0

1719-1 Only a very small (5 or 10) percentage of measurements can

1743-1 ____X___________Frequency of X

1743-2 "If the mean, median and mode of a distribution are 5, 6, 7 "

1745-1 would be most influenced by an extreme score?

1745-3 the median as 65. One would expect this distribution to be

1746-1 sensitive to extreme scores on the higher or lower end

1746-3 Which of the following is not a measure of central tendency

1797-2 "In a group of 12 scores, the largest score is increased by 36 points."

1799-2 The quantity SUM(X - XBAR) is not used as a measure of dispersion

1802-3 "In popular usage, the term average may refer to:"

1803-1 The mean is a measure of:

1805-2 X(I)______Freq(I)

1810-1 The mean XBAR of the data above is

1811-1 A sample of 5 persons with hypertension underwent a special blood

1820-3 If a teacher computes the mean for a set of test scores and then

1821-1 "In a set of 10 scores the value 2 occurs three times, the value 4"

1838-1 Prepare the frequency distribution table

1839-2 whose hot dogs would you buy? Why?

1840-1 For each characteristic obtain a frequency plot

1847-2 Which worker appears to be faster in completing the job?

1849-1 Should the manager order the study repeated to

1855-4 Is this evidence that there is likely to be a mistake

1862-1 "3,3,3,3,3 is:"

1862-2 The sample variance of the following sample of five numbers

1864-1 "The variance, S(Y)**2, of the numbers 4, 6 is"

1866-1 Let us define a new statistic as the distance between 70th sample

1867-1 Which one of the following CANNOT be used as a measure of dispersion?

1874-1 "Generally, a small standard deviation implies that the measurements"

1876-1 Which of the following relations is always correct:

1878-1 "If a constant were to be added to a set of scores, the standard"

1878-2 Increasing the frequencies in the tails of a distribution will:

1879-1 "If the variance of a distribution is 9, the standard deviation is:"

1879-2 "If 5 were subtracted from each score, the standard deviation"

1879-3 What is the relationship between the variance of a set of scores and the

1883-1 Then which of the following statements about this second sample

1884-3 "Two suppliers offer a machine part with the required 1"" diameter."

1885-3 Which of the following indices will be changed from

1886-1 "Each year, during a period of seven years, Mrs. Smith gave birth "

1887-4 "Given that n = 25, SUM(i = 1,25)((X(i) - XBAR)**2) = 600, and"

1888-2 What is the standard deviation for the following set of scores:

1889-2 Calculate the variance and standard deviataion for the data given:

1891-3 Which data set has the larger standard deviation?

1896-3 The standard deviation of a group of scores is 0 when all the scores

1897-2 The standard deviation is a point in a distribution.

1899-1 The standard deviation is a measure of dispersion around the mean.

1901-2 "If the standard deviation of a set of scores is zero, "

1907-3 "[SUM(i = 1,n)(X(i)**2)]/n is:"

1910-1 "Find S**2, the unbiased estimator"

1911-1 "The variance, S**2, of this data is closest to:"

1914-1 "The variance for the sample [47.1, 33.1, 26.1, 40.1, 54.1] is:"

1916-1 The variance of a group of 10 scores was 16. If 2 were subtracted

1920-1 The scores that have the greatest effect on the value of the variance

1920-3 Why is the numerator squared?

1950-1 The variance of a set of negative numbers is negative.

2000-1 would you expect the histogram for the group to look most like

2002-1 Explain why histograms and frequency polygons are used in statistics.

2003-1 Construct a histogram using the 9 class intervals.

2005-1 "Using your relative frequency histogram, indicate "

2006-3 Plot a histogram showing roughly the relative frequencies

2008-1 Construct a frequency distribution using the class limits:

2009-3 Explain why the above graph is misleading.

2011-1 Assume half the subjects are male and half are female.

2012-1 Discuss the problems involved in drawing a histogram from this table.

2014-1 HISTOGRAM

2015-1 "In Table A, what proportion of those whose reading speed was more than"

2016-1 and a relative frequency diagram are drawn for the following data:

2017-1 a person with a raw score of 10 has a higher score than what percent

2019-1 the total number of cases observed at each score value

2019-3 the width of each interval would be:

2020-1 "If the same test were administered to fifth graders, what"

2020-2 A frequency distribution provides the following information:

2030-1 Relative frequency distributions permit comparisons between

2033-2 A percentile score of 40 indicates that a person

2033-3 If a person earned a score higher than 35 persons in his class

2034-1 is approximately equivalent to a percentile rank of ________?

2036-1 "Using the above distribution, answer the following questions."

2037-1 "Complete this sentence: ""The kth percentile of a given distribution"

2038-1 the 78th percentile. If five points were added

2038-2 If a given score is at the 30th percentile for reference group

2042-2 "If 40% of a group obtain scores below 70, the percentile rank of"

2043-2 Half (50%) of the values in a distribution are

2044-1 "On the same test, Tom scored at the 87th percentile, and Bill scored"

2045-1 For items (i)-(iii) use the following graph.

2050-4 Explain to a friend who has not studied statistics what this fact

2053-3 The 40th percentile of a distribution is the value above which

2053-4 "If the percentile rank for the value 60 is 40, this means that"

2057-4 Frequency distributions are useful for ALL BUT which of the following

2065-1 The skewness of the population is:

2066-1 How may the distribution be described?

2119-2 A graphical presentation may accomplish ALL BUT which of the following

2890-3 A sample is:

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Q: Questions (a) to (c) refer to the following figure: | | Frequency | B | | B B | | A C | C C | | A A | C C | B B | A C A C | A C B B A C |A C B B A C ---------------------------------------------------> Score NOTE: Connect the A points with a smooth curve to form distribution A, the B points with a smooth curve to form distribution B, and the C points with a smooth curve to form distribution C. a. In the figure, which distribution's mean differs from the mean of the other two distributions? b. In the figure, which distribution has the smallest standard deviation? c. In the figure, is it likely that the mean of distribution A corresponds closely with the mode of distribution B? a. Yes b. No c. Cannot be determined

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Q: Define the following term and give an example of its use. Your example should not be one given in class or in a handout. Numerator

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Q: Define the following term and give an example of its use. Your example should not be one given in class or in a handout. Denominator

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Q: Explain briefly how you would decide which of the following two events is the more unusual: a. A 90 degree day in Vermont. b. A 100 degree day in Florida.

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Q: In what sense is the mean of any distribution the "best guess" of the score of any single individual selected at random from the group? a. In a series of such guesses, the sum of the errors in one direction will balance the sum of the errors in the other direction. b. The mean score will occur more often than any other single score. c. The chances are 50-50 that any individual will be above or below the mean. d. All the above are true. e. None of the above is true.

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Q: The heights of a sample of ten people are: 67 73 70 60 67 66 68 71 70 67. Which are the correct real limits for the frequency table given below? Frequency (a) (b) (c) 1 60.5-63.5 60-62 59.5-62.5 0 63.5-66.5 63-65 62.5-65.5 5 66.5-69.5 66-68 65.5-68.5 3 69.5-72.5 69-71 68.5-71.5 1 72.5-75.5 72-74 71.5-74.5 (1) Column a is correct (2) Column b is correct (3) Column c is correct (4) All of columns a,b,c are correct (5) None of columns a,b,c are correct

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Q: Ms. Sweetwater's biology class had a standard deviation of 2.4 on a standardized test, while Ms. Quincy's biology had a standard deviation of 1.2 on the same test. What can be said about these two classes? a. Ms. Sweetwater's class is more homogeneous than Ms. Quincy's. b. Ms. Quincy's class is less heterogeneous than Ms. Sweetwater's. c. Ms. Quincy's class did less well on the test than Ms. Sweet- water's. d. Ms. Sweetwater's class performed twice as well on the test as Ms. Quincy's.

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Q: A large mass of data can best be summarized pictorially by means of: a. the range b. a histogram c. the frequency table d. XBAR and S(Y**2)

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Q: For a symmetric distribution, the mean and median are 1. the same 2. always different 3. possibly the same, possibly different 4. insufficient information.

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Q: Consider the following data: 1, 7, 3, 3, 6, 4 The mean and median for this data are: A. 4 and 3 D. 4.8 and 3 B. 4.8 and 3 1/2 E. 4 and 3 1/2 C. 4 and 3 1/3

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Q: A distribution of 6 scores has a median of 21. If the highest score increases 3 points, the median will become ___________. a. 21 b. 21.5 c. 24 d. Cannot be determined without additional information. e. none of these

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Q: Consider the following data: 53, 61, 38, 65, 72, 58, 52, 63, 69, 74, 66 You are given that SUM(i = 1, 11)(Y(i) - YBAR)**2 = 1082 and SUM(i = 1, 11)(Y(i)) = 671. i) Find YBAR, the sample mean. a) 67.1 b) 98.4 c) 108.2 d) 61.0 e) None of the above ii) Find S**2, the sample variance. a) 98 4/11 b) 108.2 c) 0 d) 67.1 e) None of the above

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Q: If you are told a population has a mean of 25 and a variance of 0, what must you conclude? a. Someone has made a mistake. b. There is only one element in the population. c. There are no elements in the population. d. All the elements in the population are 25. e. None of the above.

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Q: True or False? If False, correct it. Only a very small (5 or 10) percentage of measurements can be more than two standard deviations from the mean.

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Q: The sample mean of the following sample X Frequency of X ----------------------------------- 2 1 3 2 4 3 is: (1) 3 (2) 2 (3) 20 (4) 20/9 = 2.22 (5) 20/6 = 3.33

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Q: If the mean, median and mode of a distribution are 5, 6, 7 respec- tively, then the distribution is: 1. skewed negatively 2. not skewed 3. skewed positively 4. symmetrical 5. bimodal.

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Q: Which of the following measures of central tendency tends to be most influenced by an extreme score? a. median b. mode c. mean

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Q: In a frequency distribution of 250 scores, the mean is reported as 78 and the median as 65. One would expect this distribution to be a. positively skewed. b. negatively skewed. c. symmetrical but not rectangular or normal. d. normal. e. rectangular.

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Q: The measure of central tendency which is sensitive to extreme scores on the higher or lower end of a distribution is the: a. median. b. mean. c. mode. d. all of the above e. none of the above

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Q: Which of the following is not a measure of central tendency? a. mean d. standard deviation b. median e. none of these c. mode

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Q: In a group of 12 scores, the largest score is increased by 36 points. What effect will this have on the mean of the scores? a. it will be increased by 12 points b. it will remain unchanged c. it will be increased by 3 points d. it will increase by 36 points e. there is no way of knowing exactly how many points the mean will be increased.

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Q: The quantity SUM(X - XBAR) is not used as a measure of dispersion because it is: (a) always equal to zero (b) always a positive value (c) too difficult to work with (d) always a negative value.

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Q: In popular usage, the term average may refer to: a. The mean b. The median c. The mode d. None of these e. All of these

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Q: The mean is a measure of: a. variability. b. position. c. skewness. d. central tendency. e. symmetry.

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Q: The mean of the following data is: X(I) Freq(I) ---------------- 0 4 1 3 2 1 3 0 4 1 a. 9/5 b. 9 c. 9/4 d. 1 e. none of the above

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Q: Consider the following data: -4, 3, 8, -2, 7, 7, 6, 11, 4, 10 The mean XBAR of the data above is (a) 4.00 (d) 5.60 (b) 4.60 (e) none of these. (c) 5.00

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Q: A sample of 5 persons with hypertension underwent a special blood- pressure-reducing treatment program which resulted in the following reductions in systolic blood pressure for these persons (i.e. the scores give SBP after treatment - SBP before treatment): -5, 10, 20, 5, 10. The mean of this sample is a. 10 b. 9 c. 8 d. 40 e. none of these

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Q: If a teacher computes the mean for a set of test scores and then subtracts this mean from each score, the SUM of the resulting set of difference scores will equal a. zero. b. unity. c. n, the number of scores. d. the mean. e. n times the mean.

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Q: In a set of 10 scores the value 2 occurs three times, the value 4 occurs twice, 6 occurs twice, and 7 occurs three times. What is the mean of the scores? a. (2 + 2 + 6 + 7)/4 b. (2 + 4 + 6 + 7)/10 c. (3*2 + 2*4 + 2*6 + 3*7)/4 d. (3*2 + 2*4 + 2*6 + 3*7)/10

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Q: The following data represent scores of 50 students in a calculus test. 72 72 93 70 59 78 74 65 73 80 57 67 72 57 83 76 74 56 68 67 74 76 79 72 61 72 73 76 67 49 71 53 67 65 100 83 69 61 72 68 65 51 75 68 75 66 77 61 64 74 a. Prepare the frequency distribution table and the frequency histogram for this data set. b. Compute the sample mean XBAR, sample median X(M), sample range R, interquantile range and sample variance S**2. c. Does the data set represent a sample or a population? If it is a sample, describe the population from which it has been drawn.

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Q: a. For each of the samples listed below obtain: 1. a mean 2. a variance, and 3. a standard deviation Each sample was randomly obtained from the production of the hot dog manufacturer listed. Company Dog Length(inches) A 5,5,5,5,5 B 6,5,5,5,4 C 9,9,5,1,1 D 9,5,5,5,1 E 9,5,5,5,5,5,5,5,5,1 F 9,9,9,4,4,3,3,3,3,3 b. Given that the price per hot dog is the same for all manufacturers, whose hot dogs would you buy? Why?

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Q: Below are measurements of characteristics for two samples of interest, For each characteristic obtain: a. Mean b. Variance c. Standard deviation d. Frequency plot DATA: MALE HEIGHTS (inches) LENGTH OF INDEX FINGER (inches) 66 71 3.0 3.2 68 71 3.5 2.8 69 72 3.4 2.9 68 69 3.5 3.1 71 71 3.7 3.1 73 70 3.2 3.3 67 72 3.5 3.2 68 69 3.4 3.5 65 70 3.1 2.9 72 70 3.5 3.1

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Q: Two workers on the same job show the following results over a long period of time. Worker Worker A B ------------------------------------------------------------------- Mean time of completing the job (minutes) 30 25 Standard deviation (minutes) 6 4 a. Which worker appears to be more consistent in the time he requires to complete the job? Explain. b. Which worker appears to be faster in completing the job? Explain.

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Q: Suppose the manager of a plant is concerned with the total number of man-hours lost due to accidents for the past 12 months. The company statistician has reported the mean number of man-hours lost per month but did not keep a record of the total sum. Should the manager order the study repeated to obtain the desired information? Explain your answer clearly.

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Q: Answer the question by circling either "YES" or "NO" and by responding to the subsequent question if it applies to your answer. Suppose I calculate the mean of a distribution to be 15 and the stan- dard deviation to be 15. Is this evidence that there is likely to be a mistake in my calculations? YES or NO? If YES, why?

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Q: The sample variance of the following sample of five numbers 3,3,3,3,3 is: (1) 0 (2) 3 (3) 9 (4) 11.25 (5) 45

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Q: The sample variance of the following sample of five numbers 1,2,3,4,5 is: (1) 2.5 (2) 9 (3) 10 (4) 13.3 (5) 55

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Q: The variance, S(Y)**2, of the numbers 4, 6 is (a) 2.0 (b) 1.0 (c) 0.5 (d) 0.25

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Q: Let us define a new statistic as the distance between 70th sample percentile and 30th sample percentile. This new statistic would give us information concerning a. central tendency. b. variability. c. relative position. d. skewness. e. symmetry.

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Q: Which one of the following CANNOT be used as a measure of dispersion? a. The difference between the third and first quartiles b. [SUM(X(i)-XBAR)]/[n] c. [SUM(]X(i)-XBAR])]/[n] d. The difference between the 90th and 10th percentiles

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Q: True or False? If False, explain why. Generally, a small standard deviation implies that the measurements are clustered close to the mean.

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Q: Which of the following relations is always correct: 1. SUM(X) = 0 2. SUM(X - XBAR) = 0 3. SUM(]X - XBAR]) = 0 4. SUM((X - XBAR)**2) = 0 5. SUM(X - a) = 0

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Q: If a constant were to be added to a set of scores, the standard deviation would: a. remain the same. b. increase by the square root of that constant. c. increase by the square of that constant. d. increase by the magnitude of that constant. e. none of the above.

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Q: Increasing the frequencies in the tails of a distribution will: a. reduce the standard deviation. b. not affect the standard deviation. c. increase the standard deviation. d. not affect the standard deviation as long as the increases are balanced on each side of the mean. e. none of the above

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Q: If the variance of a distribution is 9, the standard deviation is: a. 3 b. 6 c. 9 d. 81 e. impossible to determine without knowing n.

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Q: The standard deviation of a group of scores is 10. If 5 were subtracted from each score, the standard deviation of the new scores would be 1. 2 2. 10/25 3. 5 4. none of these.

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Q: What is the relationship between the variance of a set of scores and the standard deviation of those scores? a. variance = (standard deviation)**2 b. standard deviation = (variance)**2 c. variance = mean/standard deviation d. standard deviation = mean/variance

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Q: A sample of 5 persons with hypertension underwent a special blood- pressure-reducing treatment program which resulted in the following values giving reduction in systolic blood pressure for these per- sons (i.e. the scores give SBP after treatment - SBP before treat- ment): -5, 10, 20, 5, 10. Suppose for a second sample of 5 persons, the sample mean is 10, and the sample variance is 25. Then which of the following statements about this second sample is not correct? a. A person with a SBP reduction of 20 units is 2 standard deviations above the sample mean. b. A person with a SBP reduction of -5 units is 3 standard deviations below the sample mean. c. The sum of the squared deviations of SBP reduction scores from the sample mean, i.e. SUM((X - XBAR)**2), is 100. d. The sample median cannot be determined from the information pro- vided. e. Any SBP reduction score between 0 and 20 is within one standard deviation of the sample mean.

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Q: Two suppliers offer a machine part with the required 1" diameter. Samples indicate that supplier A's product has a standard deviation of 0.12 in. and supplier B's product has a standard deviation of 0.15 in. Which of the following statements is true? a. A offers a more homogeneous product. b. B offers a more homogeneous product. c. The standard deviation has no relationship to the quality of the product. d. Only the mean is important in buying the machine part.

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Q: The following set of scores is obtained on a test, X: 4, 6, 8, 9, 11, 13, 16, 24, 24, 24, 26. The teacher computes all of the descriptive indices of central tendency and variability on these data, then discovers that an error was made, and one of the 24's is actually a 17. Which of the following indices will be changed from the original computation? a. Median b. Mode c. Range d. Standard deviation e. None of the above

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Q: Each year, during a period of seven years, Mrs. Smith gave birth to a child. The standard deviation of the ages (in whole years) of the 7 children of the family Smith is equal to a. 2 b. 4 c. 7 d. Cannot be calculated if the present age of the children is not known

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Q: Given that n = 25, SUM(i = 1,25)((X(i) - XBAR)**2) = 600, and XBAR = 156, calculate the variance and standard deviation.

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Q: What is the standard deviation for the following set of scores: 1,1,7,13,13? In calculating the standard deviation, use the formula below: S = SQRT ((SUM(X-XBAR)**2)/(n-1))

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Q: Calculate the variance and standard deviataion for the data given: (58, 50, 55, 53, 59)

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Q: Listed below are two sample data sets, S and T. Which data set has the larger standard deviation? (Hint: you can answer this question by inspecting the two data sets. But if you are not sure after inspection, calculate the standard deviation.) Data Set S: 1, 2, 3, 4, 5, 6, 7, 8, 9 Data Set T: 8, 9, 9, 9, 10, 11, 11, 12

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Q: True or False? If false, correct it. The standard deviation of a group of scores is 0 when all the scores are the same.

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Q: True or False? If False, correct it. The standard deviation is a point in a distribution.

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Q: True or False? If False, correct it. The standard deviation is a measure of dispersion around the mean.

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Q: True or false? If false, explain why. If the standard deviation of a set of scores is zero, all the scores have the same value.

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Q: [SUM(i = 1,n)(X(i)**2)]/n is: a. the variance. b. the standard deviation. c. the variance only when the mean equals zero. d. the standard deviation only when the mean equals zero. e. none of the above.

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Q: Consider the following data: 53, 61, 38, 65, 72, 58, 52, 63, 69, 74, 66. You are given that SUM(i = 1,11)([Y(i) - YBAR]**2) = 1082 and SUM(i = 1,11)(Y(i)) = 671. Find S**2, the unbiased estimator of the sample variance. a) 98 + 4/11 b) 108.2 c) 61 d) 67.2 e) none of the above

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Q: Consider the following data: 1,7,3,3,6,4 The variance, S**2, of this data is closest to: A. 2 D. 5 B. 3 E. 6 C. 4

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Q: The variance for the sample [47.1, 33.1, 26.1, 40.1, 54.1] is: a. 72.8 d. 122.5 b. 98.0 e. none of these c. 143.1

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Q: The variance of a group of 10 scores was 16. If 2 were subtracted from each score, the variance of the new scores would be: a. 14 b. 4 c. 16 d. none of these

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Q: The scores that have the greatest effect on the value of the variance are those a. above the mean. b. below the mean. c. nearest the mean. d. farthest from the mean.

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Q: The sample variance is calculated as the average of the squared devia- tions of all the scores from the mean: SUM(i=1,n)([[X(i)-XBAR]**2]/[n-1] Why is the numerator squared? a. Because otherwise you could get a negative variance, and this is not allowed. b. Because otherwise the variance would always be equal to 0. c. Because then the square root of the variance (i.e. the standard deviation) is in the original measurement units. d. Because then the mean absolute deviation is minimal.

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Q: True or False? If false correct it. The variance of a set of negative numbers is negative.

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Q: The median weight for a group of men is 65 kg; the mean is 75 kg. Based on your knowledge of the mean and median would you expect the histogram for the group to look most like (a), (b), or (c) below? a) | | Frequency | --- | | | | | |-- | | | | | --| | |-- | | | | | | | | | | | |-- | | | | | | | | --| | | | | | | | | | | | | | | | | | | | | |-- | | | | | | | | | | | | | | | | | |-------- | | | | | | | | | | | | | | | | | | | | | | | | | |-------- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | --//-----------+---------------------------> 0 65 Weight in Kg b) | | Frequency | | | --- | | | | | | | --| |-- | | | | | | --| | | |-- | --| | | | | |-- | | | | | | | | | | --| | | | | | | |-- | | | | | | | | | | | | | | | | | | | | | | --//-----------+---------------------------> 0 65 Weight in Kg c) | | Frequency | | | --- | | | | --| | | | | | | --| | |-- | | | | | | | --| | | | | | | | | | | | | | | | | | | | ----| | | | | | | | | | | | | | | | ----| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | --//-----------+---------------------------> 0 65 Weight in Kg

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Q: Explain why histograms and frequency polygons are used in statistics.

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Q: On a final examination, the following scores were earned: 5,6,7,7, 10,15,16,16,17,17,22,22,22,25,26,28,31,33,35,37,40. Use these data to answer the following 4 questions. 1. Construct a frequency table for this data, grouping the data into 9 class intervals. 2. Construct a histogram using the 9 class intervals. Be sure to properly construct and label the histogram. 3. What are the exact or real limits of the lowest class interval?

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Q: Suppose that the variable Y can only take on the following five values with relative frequencies as indicated below: Y Rel. Freq. Cumulative Rel. Freq. - ---------- --------------------- 0 5/100 1 25/100 2 30/100 3 25/100 4 15/100 a. Fill in the cumulative relative frequencies. b. Using 5 classes with unit widths and identifying the midpoints of the classes with the values of Y, plot the relative frequency histogram for the data. c. Using your relative frequency histogram, indicate the positions of the mean, median, and mode. d. If relative frequency is given as the interpretation of probability, then what is the probability that Y is greater than or equal to 2?

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Q: A report states that a measurement is approximately normally distributed with mean 3.5 and variance 1. Further, it states that measurements were recorded for 7 measurement classes 0-1,1-2, etc. Plot a histogram showing roughly the relative frequencies of these measurement classes.

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Q: Given the following data: 18, 13, 2, 20, 8, 10, 5, 10, 6, 9, 10, 20, 2 15, 16, 16, 13, 10, 17, 10, 3, 2, 15, 8, 5 a) Construct a frequency distribution using the class limits: 1-4, 5-8, 9-12, 13-16, 17-20. b) Construct a relative frequency histogram using the results of part a). c) Why might it be useful to construct a frequency distribution and/or a histogram of the sample data?

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Q: | | 150 + ----- | | | 100 + ----- | | # OF | | | | | PEOPLE 30 + | | | | ----- ----- | | | | | | | | | 5 + | | | | | | | | ----- | | | | | | | | | | | --------+-------+-------+-------+-------+----- 1 2 3 4 5 or less or more AGE OF CAR IN YEARS Explain why the above graph is misleading.

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Q: Suppose you are given a data set to analyze. The data consist of 1000 observations on one variable, the height of the subject being inter- viewed. Assume half the subjects are male and half are female. The sample mean height of the males is larger than the sample mean height of the females, while the sample standard deviation of the females is larger than the sample standard deviation of the males. Both histo- grams are approximately symmetric. Draw two histograms, one for males and one for females, to illustrate these facts.

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Q: A friend of yours heard that you were taking statistics and has presented you with the following table from which he wants you to construct a histogram. Age Relative Frequency (%) ----- ---------------------- 0-14 28.4 15-44 50.5 45+ 21.1 ----- 100.0 Discuss the problems involved in drawing a histogram from this table.

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Q: Define the following term and give an example of its use. Your example should not be one given in class or in a handout. HISTOGRAM

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Q: TABLE A The following table is a cross-tabulation of age and reading speed of 100 pupils. Age to the nearest month 94-103 104-113 114-123 124-133 134-143 144-153 | f(Y) | 46-51 4 1 | 5 Score 40-45 2 10 5 | 17 on 34-39 1 5 2 | 8 reading 28-33 2 8 6 3 1 1 | 21 speed 22-27 9 13 3 | 25 16-21 1 5 4 2 1 1 | 14 10-15 2 3 3 1 1 | 10 --------------------------------------------------------------- f(X) 6 43 34 11 3 3 |100 In Table A, what proportion of those whose reading speed was more than 33 were aged between 104 and 113 months? a. 19/30 b. 27/51 c. 19/43 d. 27/43 e. 19/100

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Q: Both a frequency diagram and a relative frequency diagram are drawn for the following data: X f 15 3 17 5 19 4 23 3 29 1 We can state that a. both will have the same values on the horizontal axis. b. the frequency diagram will be less precise than the relative frequency diagram. c. one depicts data in many-value classes while the other depicts single-value classes. d. both depict continuous variables.

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Q: The figure below indicates that a person with a raw score of 10 has a higher score than what percent of the persons in the group? | NOTE: In order to complete the figure, | connect the *'s with a smooth curve. Cumulative | Proportion 100 + * | 90 + * | * 80 + | 70 + | 60 + | 50 + | 40 + * | 30 + | 20 + | * 10 + * | * *---+---+---+---+---+---+---+---+--------> 2 4 6 8 10 12 14 16 Raw Score a. 10 b. 50 c. 82 d. 93

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Q: A list of the percentages of the total number of cases observed at each score value or each subinterval of scores is: a. a histogram d. a normal distribution b. a relative frequency distribution e. none of these c. a cumulative frequency polygon

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Q: If our lowest score were 40, and the highest score were 189, n=200, and we decide to group our scores into 15 class intervals for a frequency distribution, the width of each interval would be: a. 5 b. 10 c. 12 d. 15 e. Insufficient information to answer

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Q: A reading test with 50 possible points yields a bell-shaped distribution with scores ranging from 5 to 48 on a large sample of third graders. If the same test were administered to fifth graders, what would we expect the form of the frequency distribution to be? a. Negatively skewed (skewed to the left) b. Symmetric and bell-shaped c. Symmetric, but not bell-shaped d. Positively skewed (skewed to the right) e. None of the above

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Q: A frequency distribution provides the following information: a) The value of the measurement and the number of individuals with that value. b) The value of the measurement and the percent of individuals with that value. c) The value of the measurement and the percent of individuals with that value or a smaller one.

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Q: True or False? If False, correct it. Relative frequency distributions permit comparisons between data sets of different sizes.

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Q: A percentile score of 40 indicates that a person a. answered 40% of the questions correctly on the test; b. knows 40% of the material covered by the examination; c. has earned a score equal to or better than 40 persons in his class; d. has earned a score equal to or better than 40% of the persons in his class.

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Q: If a person earned a score higher than 35 persons in his class of 50 students, what is his percentile score? a. 35 b. 50 c. 70 d. 90

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Q: The scores on a midterm examination are presented below in decreasing order of magnitude. A score of 63 is approximately equivalent to a percentile rank of ________? 65 63 61 60 57 56 65 62 61 59 57 56 65 62 60 59 57 56 64 62 60 58 57 48 64 61 60 58 57 47 a. 20 b. 25 c. 63 d. 80

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Q: Class interval f Cum f Cum % 75-83 5 200 100.0 66-74 12 195 97.5 57-65 15 183 91.5 48-56 38 168 84.0 39-47 60 130 65.0 30-38 40 90 35.0 21-29 13 30 15.0 12-20 10 17 8.5 3-11 7 7 3.5 ------- N=200 Using the above distribution, answer the following questions. 1) The frequency of 38 in the interval 48-56 means: a. 38 frequencies are at the upper real limit of the interval. b. 38 frequencies are at the lower real limit of the interval. c. 38 frequencies are spread out throughout the interval. d. 38 frequencies are at the upper apparent limit of the interval. e. 38 frequencies are at the lower apparent limit of the interval. 2) A cumulative percentage of 97.5 means that: a. 97.5 cases fall below a score of 74. b. 97.5% of the cases fall below a score of 74. c. 97.5% of the cases fall below a score of 65.5. d. 97.5% of the cases fall below a midpoint of the interval 66-74. e. 97.5% of the cases fall below a score of 74.5. 3) The score above which 35% of the cases are found is: a. 70 b. 40 c. 38.5 d. 29.5 e. 47.5

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Q: Complete this sentence: "The kth percentile of a given distribution is ...": a. the score at which k% of the cases fall. b. the score above which k% of the cases fall. c. the score below which k% of the cases fall. d. cannot answer without knowing the precise numerical value of k. e. more than one of the above is correct.

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Q: Edith G. obtained a score of 65 in a statistics test, placing her at the 78th percentile. If five points were added to each score in the distribution, her new score would be at the: a. 83rd percentile. b. 78th percentile. c. 70th percentile. d. none of the above. e. impossible to answer without additional information.

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Q: If a given score is at the 30th percentile for reference group A and the 60th percentile for reference group B, which of the following is most likely true? a. Individuals in reference group B generally performed better on the test than those in group A. b. A person at the 15th percentile with group A will be at the 30th percentile with group B. c. A person at the 80th percentile with reference group B will be at the 50th percentile with group A. d. Individuals in reference group B generally scored lower on the test than those in reference group A. e. None of the above.

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Q: If 40% of a group obtain scores below 70, the percentile rank of the score is: 1. 30 4. 70 2. 40 5. none of these 3. 60

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Q: The following data are the number of hours worked per week by seven State College students: 3, 7, 4, 6, 2, 8, 19 Half (50%) of the values in a distribution are a. included in the range c. between the mean and mode b. between Q(1) and Q(3) d. the mode and the highest value

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Q: On the same test, Tom scored at the 87th percentile, and Bill scored at the 73rd. This means a. Tom is 14% better than Bill. b. Tom scored 14 more points than Bill. c. 14% of those taking the test got scores ranging between Tom's and Bill's. d. there were only 13 people smarter than both Tom and Bill. e. none of the above.

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Q: For items (i)-(iii) use the following graph. Ogives of Distributions of Arithmetic Test Scores for Seventh and Eighth Graders (Note: Connect 7's with a smooth curve for seventh grade ogive and connect 8's with a smooth curve for eighth grade ogive.) 100+------------------------------------------------7---------8 | | | | | | | - | | | | 7 | 8 | | | | | | | | 90+--------|---------|---------|---------|--7------|-----8---| | | | | | | | - | | | 7 | | | | | | | | | 80+--------|---------|---------|-------7-|---------|---8-----| | | | | | | | - | | | 7 | | | | | | | | | | 70+--------|---------|---------|---------|---------|-8-------| | | | | | | | - | | | | | | | | | | | 8 | PERCENT 60+--------|---------|---------|---------|---------|---------| | | | | | | | - | | | | | | | | | | | | | 50+--------|---------|---------|---------|---------|---------| | | | | | | | - | | 7 | 8 | | | | | | | | | 40+--------|---------|---------|---------|---------|---------| | | | | | | | - | | | | | | | | | | | | | 30+--------|---------|-----7---|---------|--8------|---------| | | | | | | | - | | | 8 | | | | | | | | | 20+--------|---------|-7-------|-------8-|---------|---------| | | | | | | | - | 7 | | 8 | | | | | | | | | | 10+--------|---------|-------8-|---------|---------|---------| | | | | | | | - 7 8 | | | | | | | | | | | 0+----7---|--8------|---------|---------|---------|---------| 5 10 15 20 25 30 TEST SCORE (i) What is the estimated percent of 8th grade pupils whose arith- metic scores fall below the median score for grade 7? a. 6 b. 12 c. 16 d. 24 e. It is impossible to estimate this percent from the ogives. (ii) What would be the shape of the frequency distribution corres- ponding to the 8th grade ogive? a. Skewed to the left b. Skewed to the right c. Bimodal d. Unimodal and symmetrical e. It is impossible to tell from the ogive.

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Q: Suppose that the 60th percentile of a sample was 1468.3. Explain to a friend who has not studied statistics what this fact tells you about this sample.

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Q: True or False? If false, correct it. The 40th percentile of a distribution is the value above which 60% of the distribution of values will fall.

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Q: True or False? If false, correct it. If the percentile rank for the value 60 is 40, this means that 60% of the observations will fall above 40.

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Q: Frequency distributions are useful for ALL BUT which of the following objectives? a) Investigation of characteristics of each observation. b) Summarization of data. c) Condensation of large sets into smaller sets. d) Illustration of the amount of variability in data.

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Q: The mean of the population of ten scores, 78, 91, 91, 94, 74, 23, 63, 22, 78, 89 is 70.3, and the modes are 78 and 91. The skewness of the population is: 1. negative 2. zero 3. positive 4. not determined 5. positive or negative depending on the score.

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Q: The distribution of entrance test scores of freshmen in a particular university has the following percentile scores. How may the distribution be described? Percentile Score ---------- ----- 95th 140 80th 120 65th 101 50th 94 35th 91 20th 87 5th 80 a. Symmetrical bell-shaped b. Skewed left (negatively skewed) c. Skewed right (positively skewed) d. Impossible to tell from the above

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Q: A graphical presentation may accomplish ALL BUT which of the following objectives? a) Illustrate the amount of variation in the data. b) Illustrate approximately where the mean is. c) Allow comparison with similar data. d) Will have the exact same shape regardless of what units are used on the axes.

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Q: A sample is: a. a number resulting from the manipulation of raw data according to specified rules. b. a subset of a population. c. a characteristic of a population which is measurable. d. a complete set of individuals, objects, or measurements having some common observable characteristic. e. none of the above.

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A: a. The mean of distribution C differs from the means of distributions A and B. b. Distribution B has the smallest standard deviation. c. Yes, it is likely that the mean of distribution A corresponds closely with the mode of distribution B.

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A: Definition: The value divided when division is carried out. Example: The expression 12/5 or (12 divided by 5) indicates that 12 is to be divided by 5. 12 is the numerator.

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A: Definition: The value that "goes into" the value divided when division is carried out or the value below the division symbol in a fraction. Example: The expression 12/5 or (12 divided by 5) indicates that 12 is to be divided by 5. 5 is the denominator. It is the number that "goes into" 12.

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A: One would examine previous weather records and note the relative fre- quency of 90+ degree days in Vermont, and 100+ degree days in Florida. Thus, relative frequencies are one method of estimating the probability of each event. (Note: The smallest frequency is the more unusual event.)

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A: a. In a series of such guesses, the sum of the errors in one direction will balance the sum of the errors in the other direction. Answers b. and c. are not necessarily true with distributions other than normal distributions.

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A: (3) Column c is correct

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A: b. Ms. Quincy's class is less heterogeneous than Ms. Sweetwater's.

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A: b. a histogram

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A: 1. the same

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A: E. 4 and 3.5 Mean = (1+7+3+3+6+4)/6 = 4 Data arranged in ascending order: 1,3,3,4,6,7 Median = (3+4)/2 = 3.5

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A: a. 21 Median is not sensitive to the relative magnitude of the scores but is determined by their rank ordering, which is not affected by this change.

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A: i) 671/11 = 61.0, (d) ii) 1082/10 = 108.2, (b)

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A: d. All the elements in the population are 25.

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A: False. There are many distributions for which this statement is false, however it is true for distributions that are approximately normal.

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A: (5) 20/6 = 3.33

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A: 1. skewed negatively.

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A: c. mean

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A: a. positively skewed. In a positively skewed distribution the mean is greater than the median, while in a negatively skewed distribution the mean is less than the median, and in a symmetrical distribution the mean is equal to the median.

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A: b. mean.

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A: d. standard deviation Standard deviation is a measure of dispersion among the data.

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A: c. it will be increased by 3 points

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A: (a) always equal to zero.

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A: (E.) It may refer to any of these.

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A: d. central tendency.

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A: d. 1 mean = (SUM(X(I)*Freq(I)))/SUM(Freq(I))

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A: (c) 5.00

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A: c. 8 (-5 + 10 + 20 + 5 + 10)/5 = 8

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A: a. zero.

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A: d. (3*2 + 2*4 + 2*6 + 3*7)/10

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A: a. SCORE | FREQUENCY ------------------------------ 90 - 100 | 2 80 - 89 | 3 70 - 79 | 22 60 - 69 | 16 50 - 59 | 6 40 - 49 | 1 ----- | | | | -----| | | | | | | | | | | -----| | | | | | | | | | |----- | | | | | -----| | | | |----- | | | | | | | | | | | | | ----------------------------------------------- 45 55 65 75 85 95 b. XBAR = 69.94 Median = 77.5 Interquantile Range = 75 - 65 = 10 Variance = 91.65 c. Insufficient information is given, but one assumes from part (b) that it is a sample.

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A: a. Company Mean Variance St. Dev. (SUM X(i))/n=XBAR S**2=(SUM(X-XBAR)**2)/n-1 S=SQRT(VAR.) A 25/5=5 0 0 B 25/5=5 2/4=1/2=.5 SQRT( 1/2 )=.707 C 25/5=5 64/4=16 SQRT( 16 )=4 D 25/5=5 32/4=8 SQRT(8)=2.83 E 50/10=5 32/9=3.55 SQRT( 3.55)=1.89 F 50/10=5 70/9=7.78 SQRT(7.78)=2.79 b. This question may have a variety of answers. The decision would depend on the purpose. If it was important to have as little variability as possible when selling 5 inch hot dogs, company A would be best since it has the least variability. However, if you could profit from selling hot dogs in a variety of lengths, company F might prove best since it shows a lot of variability and produces hot dogs ranging from 9 to 3 inches in length.

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A: a. XBAR(height) = 1392/20 = 69.6 XBAR(length) = 64.9/20 = 3.245 b. S**2(height) = (SUM(X - 69.6)**2)/19 = 12.96 1.96 2.56 1.96 .36 5.76 deviations 2.56 .36 from the 1.96 1.96 mean squared 11.56 .16 are: 6.76 5.76 2.56 .36 21.16 .16 5.76 .16 S**2(height) = 86.8/19 = 4.568 S**2(length) = (SUM(X - 3.245)**2)/19 = .06 .002 .065 .198 .024 .119 deviations .065 .021 from the .207 .021 mean squared .002 .003 are: .065 .002 .024 .065 .021 .119 .065 .021 S**2(length) = 1.169/19 = .06 c. S(height) = SQRT(4.568) = 2.137 S(length) = SQRT(.06) = .25 d. FREQUENCY PLOTS: | | MALE HEIGHTS | | 6 + Frequency | 5 + | 4 + x | 3 + x x x x | 2 + | 1 + x x x x | -----+----+----+----+----+----+----+----+----+----+--> 65 66 67 68 69 70 71 72 73 74 Inches | | | LENGTH OF INDEX FINGER | 6 + Frequency | 5 + x | 4 + x | 3 + x | 2 + x x | 1 + x x x x | -----+----+----+----+----+----+----+----+----x----+--> 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Inches NOTE: To complete frequency plots, connect x's with straight lines.

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A: a. Worker B appears to be more consistent in the time he requires to complete the job, since he has a smaller variance. b. Worker B appears to be faster in completing the job, since he has a smaller mean. (You could actually test this.)

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A: No--the estimate that he would get using the mean number per month would most likely be accurate enough, without having to go to the extra expense of another study. Presumably the mean number of hours lost per month is equal to the total number of hours lost divided by 12, so it's not difficult to calculate the total.

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A: NO, because mean and standard deviation may not specify the distribu- tions completely. Even if they do so, (as in normal distributions) there is no relation between mean and standard deviation. Hence, if there is a calculation mistake, it is not detectable by the given in- formation. In fact, for the exponential distribution, the mean exactly equals the standard deviation.

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A: (1) 0

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A: (1) 2.5

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A: (a) 2.0 SS = 2.0 and n - 1 = 1 S(Y)**2 = 2.0/1 = 2.0

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A: b. variability.

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A: b. [SUM(X(i)-XBAR)]/[n] SUM(X(i)-XBAR) always equals zero.

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A: True

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A: 2. SUM(X - XBAR) = 0

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A: a. remain the same.

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A: c. increase the standard deviation.

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A: a. 3 The variance is equal to the standard deviation squared.

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A: 4. none of these Correct answer: 10.

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A: a. variance = (standard deviation)**2

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A: e. Any SBP reduction score between 0 and 20 is within one standard deviation of the sample mean. 10 +/- SQRT(25) is the interval from 5 to 15.

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A: a. A offers a more homogeneous product.

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A: d. Standard deviation

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A: a. 2 For the calculation of the standard deviation it does not make any difference if the present ages are 1,2,3,4,5,6, and 7 years or 15, 16,17,18,19,20, and 21, or whatsoever. So take 1,2,3,4,5,6, and 7 and calculated S = SQRT([[X-MU]**2]/[n]).

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A: Variance = 600/24 = 25 Standard deviation = SQRT(25) = 5

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A: XBAR = (1+1+7+13+13)/5 = 7 TOTAL X 1 1 7 13 13 35 X-XBAR -6 -6 0 6 6 0 (X-XBAR)**2 36 36 0 36 36 144 S= SQRT (144/4) = SQRT (36) = 6

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A: Variance = (SUM((X - XBAR)**2))/(n - 1) = 13.5 Standard Deviation = SQRT[SUM((X - XBAR)**2)/(n - 1)] == 3.67

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A: On inspection, data set S has the larger sample standard deviation. By calculation: S(S) = 2.58 S(T) = 1.27

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A: True.

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A: False. The standard deviation is a measure of spread in a distribution.

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A: True.

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A: True.

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A: c. the variance only when the mean equals zero.

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A: b) 108.2 Sample Variance = (1/(N-1))SUM(i=1,11)([Y(i)-YBAR]**2) = (1/(11 - 1))(1082) = (1/10)(1082) = 108.2

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A: D. 5 S**2 = SUM[(X(i) - XBAR)**2/(n-1)], where XBAR is the mean of the data = (9+9+1+1+4+0)/5 = 24/5 = 4.8

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A: d. 122.5 Variance = [SUM(i=1,n)((X(i) - XBAR)**2)]/[n - 1] = ([(47.1-40.1)**2] + [(33.1-40.1)**2] + [(26.1-40.1)**2] + [(40.1-40.1)**2] + [(54.1-40.1)**2])/[4] = 490/4 = 122.5

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A: c. 16

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A: d. farthest from the mean.

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A: b. Because otherwise the variance would always be equal to 0. Note: SUM(i=1,n)(X-XBAR) = 0

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A: False. The variance of a set of negative numbers is positive.

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A: a. The mean is greater than the median, indicating the histogram would be skewed with its tail to the right.

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A: 1) A picture is worth a thousand words, i.e. a visual display is a good way to summarize a large data set. 2) Histograms give some feeling for the overall shape of the data, something that reporting XBAR and S cannot give, i.e. skewness, kurtosis, range. 3) Relative histograms and frequency polygons are estimates of proba- bility functions.

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A: 1. FREQUENCY TABLE Score Midpoint Frequency 37-40 38.5 2 33-36 34.5 2 29-32 30.5 1 25-28 26.5 3 21-24 22.5 3 17-20 18.5 2 13-16 14.5 3 9-12 10.5 1 5-8 6.5 4 1-4 2.5 0 2. HISTOGRAM Frequency | | | 5 + | | 4 + _____ | | | | | | 3 + | | _____ _________ | | | | | | | | | | | | | | | | 2 + | | | |___| | | _________ | | | | | | | | | | | | | | | | | | | | | | 1 + | |___| | | | |___| | | | | | | | | | | | | | | | | | | | | | | | | -+---+---+---+---+---+---+---+---+---+---+--------> .5 8.5 16.5 24.5 32.5 40.5 4.5 12.5 20.5 28.5 36.5 Final Examination Scores 3. The real or exact limits of the lowest interval are 4.5 - 8.5.

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A: a. Y Cumulative Rel. Freq. - --------------------- 0 5/100 1 30/100 2 60/100 3 85/100 4 100/100 b. | | 30 + ----- | | | + ----| |---- | | | | | 20 + | | | | Relative | | | | | Frequency + | | | |---- | | | | | | 10 + | | | | | | | | | | | +---| | | | | | | | | | | --+---+---+---+---+--------> Y 0 1 2 3 4 c. Median = 1.5 + (20/30)(1.0) = 2.17 Mode = 2 Mean = [0(5) + 1(25) + 2(30) + 3(25) + 4(15)]/100 = 220/100 = 2.2 d. Prob(Y >= 2) = (30/100) + (25/100) + (15/100) = 70/100 = .70 = 70%

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A: .4 x .3 x x .2 .1 x x 0 x x _____________________________________ 0-1 1-2 2-3 3-4 4-5 5-6 6-7

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A: a) Class Freq. Rel. Freq. 1-4 4 4/25 5-8 5 5/25 9-12 6 6/25 13-16 6 6/25 17-20 4 4/25 Totals 25 25/25 b) | | 10/25 + | Relative | Frequency | | ----------------- 5/25 + --------| | | |--------| | | |-------- || | | | | | || | | | | | || | | | | | --+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-----> 1 4 5 8 9 12 13 16 17 20 X Values c) 1. Sometimes it is easier to get the mean and variance (group method -- by hand). 2. Visual check on calculation of mean and variance. 3. Easier for lay person (specialist also) to see what values are likely and what values are unlikely in a population.

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A: The primary problem is that the vertical axis does not have equally spaced intervals. For example, a constant distance on the axis re- presents jumps of 5, 25, and 70 units.

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A: | | FEMALES Frequency | | ----- | ----| |---- | ----| | | |---- | ----| | | | | |---- | ----| | | | | | | |---- | | | | | | | | | | | | | | | | | | | | | | -----------------------+---------------------------> XBAR | | MALES Frequency | | | ----- | | | | | | | | | | ----| |---- | | | | | | | | | | | ----| | | |---- | | | | | | | | | | | | | | | | | | | | | | | | | | | | ------------------------------------+--------------> YBAR Histogram for males should be taller and thinner and shifted to the right of that for females. The areas of the two histograms should be the same.

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A: 1. Too few intervals to give meaningful results. 2. Open ended interval 3. Intervals of unequal length.

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A: Definition: A graph indicating frequencies or relative frequencies of values (or value classes) of a random variable. Example: Suppose that we interviewed 10 sultans to discover how many wives each sultan enjoyed. Suppose that the results were: 1 sultan had no wives 4 sultans had two wives 3 sultans had three wives, and 2 sultans had four wives. Then the appropriate histogram would be: | Number of | Sultans | | 4 + ------- | | | 3 + | |------ | | | | 2 + | | |------ | | | | | 1 +------ | | | | | | | | | | ---+-----+-----+-----+-----+---------> 0 1 2 3 4 Number of Wives

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A: a. 19/30 30 students scored more than 33(5+17+8 = 30) of which 19(4+10+5) were in the age range 104-113 months.

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A: a. both will have the same values on the horizontal axis.

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A: c. 82

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A: b. a relative frequency distribution.

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A: b. 10; (190-40)/15 = 10

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A: a. Negatively skewed (skewed to the left), because they score higher in general.

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A: a) The value of the measurement and the number of individuals with that value.

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A: True.

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A: d. has earned a score equal to or better than 40% of the persons in his class.

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A: c. 70

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A: (d)

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A: 1) c. 38 frequencies are spread out throughout the interval. 2) e. 97.5% of the cases fall below a score of 74.5. 3) e. 47.5

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A: c. the score below which k% of the cases fall.

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A: b. 78th percentile.

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A: d. Individuals in reference group B generally scored lower on the test than those in reference group A.

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A: 2. 40

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A: b. between Q(1) and Q(3)

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A: c. 14% of those taking the test got scores ranging between Tom's and Bill's.

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A: (i) b. 12 (ii) a. skewed to the left

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A: It tells you that 60% of the sample has values less than or equal to 1468.3.

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A: True.

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A: False. If the percentile rank for the value 60 is 40, this means that 60% of the observations will fall above 60.

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A: a) Investigation of characteristics of each observation.

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A: 1. negative, because the mean is below the median. The two low values, (22, 23), are primarily responsible.

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A: c. Skewed right (positively skewed)

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A: d) Will have the exact same shape regardless of what units are used on the axes.

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A: b. a subset of a population.

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