Multiple Choice Questions Descriptive Statistics - Summary Statistics 1. Last year a small statistical consulting company paid each of its five statistical clerks $22,000, two statistical analysts $50,000 each, and the senior statistician/owner $270,000. The number of employees earning less than the mean salary is: (a) 0 (b) 4 (c) 5 (d) 6 (e) 7 2. The following table represents the relative frequency of accidents per day in a city. Accidents 0 1 2 3 4 or more Relative 0.55 0.20 0.10 0.15 0 Frequency Which of the following statements are true? I. The mean and modal number of accidents are equal. II. The mean and median number of accidents are equal. III. The median and modal number of accidents are equal. (a) I only (b) II only (c) III only (d) I, II and III (e) I, II 1
3. During the past few months, major league baseball players were in the process of negotiating with the team owners for higher minimum salaries and more fringe benefits. At the time of the negotiations, most of the major league baseball players had salaries in the $100,000 ů $150,000 a year range. However, there were a handful of players who, via the free agent system, earned nearly three million dollars per year. Which measure of central tendency of players salaries, the mean or the median, might the players have used in an attempt to convince the team owners that they (the players) were deserving of higher salaries and more fringe benefits? (a) Not enough information is given to answer the question. (b) Either one, because all measures of central tendency are basically the same. (c) Mean. (d) Median. (e) Both the mean and the median. 4. A financial analyst s sample of six companies book value were $25, $7, $22, $33, $18, $15. The sample mean and sample standard deviation are (approximately): (a) 20 and 79.2 respectively (b) 20 and 8.9 respectively. (c) 120 and 79.2 respectively. (d) 20 and 8.2 respectively. (e) 120 and 8.9 respectively. 5. A sample of underweight babies was fed a special diet and the following weight gains (lbs) were observed at the end of three month. 6.7 2.7 2.5 3.6 3.4 4.1 4.8 5.9 8.3 The mean and standard deviation are: (a) 4.67, 3.82 (b) 3.82, 4.67 (c) 4.67, 1.95 (d) 1.95, 4.67 (e) 4.67, 1.84 c 2006 Carl James Schwarz 2
6. The effect of acid rain upon the yield of crops is of concern in many places. In order to determine baseline yields, a sample of 13 fields was selected, and the yield of barley (g/400 m 2 ) was determined. The output from SAS appears below: QUANTILES(DEF=4) EXTREMES N 13 SUM WGTS 13 100% MAX 392 99% 392 LOW HIGH MEAN 220.231 SUM 2863 75% Q3 234 95% 392 161 225 STD DEV 58.5721 VAR 3430.69 50% MED 221 90% 330 168 232 SKEW 2.21591 KURT 6.61979 25% Q1 174 10% 163 169 236 USS 671689 CSS 41168.3 0% MIN 161 5% 161 179 239 CV 26.5958 STD MEAN 16.245 1% 161 205 392 The mean, standard deviation, median, and the highest value are: (a) 220.231 3430.60 50% 225 (b) 220.231 16.245 221 225 (c) 220.231 58.5721 50% 392 (d) 220.231 58.5721 221 392 (e) 220.231 58.5721 234 392 7. The effect of salinity upon the growth of grasses is of concern in many places where excess irrigation is causing salt to rise to the surface. In order to determine baseline yields, a sample of 24 fields was selected, and the biomass of grasses in a standard sized plot was measured (kg). The output from SAS appears below: QUANTILES(DEF=4) EXTREMES N 24 SUM WGTS 24 100% MAX 22.6 99% 22.6 LOW HIGH MEAN 9.09 SUM 218.3 75% Q3 11.45 95% 22.52 0.7 15.1 STD DEV 6.64 VARIANCE 44.0 50% MED 8.15 90% 21.8 1 19.8 SKEWNE 0.924 KURTO -0.0209 25% Q1 3.775 10% 1.6 2.2 21.3 USS 2998 CSS 1012.73 0% MIN 0.7 5% 0.77 2.2 22.3 CV 72 STD MEAN 1.35 1% 0.7 2.8 22.6 T:MEAN=0 6.7153 PROb> T 0.0001 RANGE 21.9 SGN RANK 150 PROb> S 0.0001 Q3-Q1 7.675 The mean, standard deviation, tenth percentile, and the highest value are: (a) 9.09 44.0 10% 22.6 (b) 9.09 6.64 1.6 15.1 (c) 9.09 6.64 21.8 15.1 (d) 9.09 6.64 1.6 22.6 (e) 9.09 1.35 21.8 15.1 c 2006 Carl James Schwarz 3
8. The heights in centimeters of 5 students are: 165, 175, 176, 159, 170. The sample median and sample mean are respectively: (a) 170, 169 (b) 170, 170 (c) 169, 170 (d) 176, 169 (e) 176, 176 9. If most of the measurements in a large data set are of approximately the same magnitude except for a few measurements that are quite a bit larger, how would the mean and median of the data set compare and what shape would a histogram of the data set have? (a) The mean would be smaller than the median and the histogram would be skewed with a long left tail. (b) The mean would be larger than the median and the histogram would be skewed with a long right tail. (c) The mean would be larger than the median and the histogram would be skewed with a long left tail. (d) The mean would be smaller than the median and the histogram would be skewed with a long right tail. (e) The mean would be equal to the median and the histogram would be symmetrical. 10. In measuring the centre of the data from a skewed distribution, the median would be preferred over the mean for most purposes because: (a) the median is the most frequent number while the mean is most likely (b) the mean may be too heavily influenced by the larger observations and this gives too high an indication of the centre (c) the median is less than the mean and smaller numbers are always appropriate for the centre (d) the mean measures the spread in the data (e) the median measures the arithmetic average of the data excluding outliers. 11. In general, which of the following statements is FALSE? (a) The sample mean is more sensitive to extreme values than the median. c 2006 Carl James Schwarz 4
(b) The sample range is more sensitive to extreme values than the standard deviation. (c) The sample standard deviation is a measure of spread around the sample mean. (d) The sample standard deviation is a measure of central tendency around the median. (e) If a distribution is symmetric, then the mean will be equal to the median. 12. The frequency distribution of the amount of rainfall in December in a certain region for a period of 30 years is given below: Rainfall Number (in inches) of years 2.0-4.0 3 4.0-6.0 6 6.0-8.0 8 8.0-10.0 8 10.0-12.0 5 The mean amount of rainfall in inches is: (a) 7.30 (b) 7.25 (c) 7.40 (d) 8.40 (e) 6.50 13. A consumer affairs agency wants to check the average weight of a new product on the market. A random sample of 25 items of the product was taken and the weights (in grams) of these items were classified as follows: Class Limits Frequency 74-77 3 77-80 6 80-83 9 83-86 3 86-89 4 The 3rd quartile of the weight in this sample is equal to: (a) 83.00 (b) 75.00 (c) 83.75 c 2006 Carl James Schwarz 5
(d) 18.75 (e) 84.50 14. A random sample of 40 smoking people is classified in the following table: Ages Frequency 10-20 4 20-30 6 30-40 12 40-50 10 50-60 8 Total 40 The mean age of this group of people. (a) 4.5 (b) 8.0 (c) 34.5 (d) 38.0 (e) 1520.0 15. A frequency distribution of weekly wages for a group of employees is given below: Weekly wages Frequency 50.00-75.00 10 75.00-100.00 15 100.00-125.00 60 125.00-150.00 40 150.00-175.00 10 The mean for this group is: (a) $112.50 (b) $125.00 (c) $105.41 (d) $117.13 (e) $118.50 16. Consider the following cumulative relative frequency distribution: Less than or equal to Cum. rel. freq. 5.0 0.23 10.0 0.34 15.0 0.41 20.0 1.00 c 2006 Carl James Schwarz 6
If this distribution is based on 800 observations, then the frequency in the second interval is: (a) 34 (b) 272 (c) 80 (d) 88 (e) 456 The following information will be used in the next three questions. A sample of 35 observations were classified as follows: Class Frequency 0-5 8 5-10 2 10-15 6 15-20 8 20-25 5 25-30 5 30-35 0 35-40 1 17. The class mark of the third class is: (a) 10.0 (b) 12.5 (c) 15.0 (d) 7.5 (e) 17.5 18. The sample mean of the above grouped data is: (a) 14.89 (b) 14.23 (c) 15.35 (d) 15.11 (e) 14.74 19. The 80th percentile of the above grouped data is: (a) 27 (b) 22 c 2006 Carl James Schwarz 7
(c) 19 (d) 23 (e) 24 20. Recently, the City of Winnipeg has been criticized for its excessive discharges of untreated sewage into the Red River. A microbiologist take 45 samples of water downstream from the treated sewage outlet and measures the number of coliform bacteria present. A summary table is as follows: Number of Number of Bacteria Samples 20-30 5 30-40 20 40-50 15 50-60 5 The 80th percentile is approximately: (a) 45 (b) 47 (c) 80 (d) 48 (e) 36 21. Recently, the City of Winnipeg has been criticized for its excessive discharges of untreated sewage into the Red River. A microbiologist take 50 samples of water downstream from the treated sewage outlet and measures the number of coliform bacteria present. A summary table is as follows: Number of Number of Bacteria Samples 50-60 5 60-70 20 70-80 10 80-90 15 The mean number of bacteria per sample is: (a) 70 (b) 71 (c) 72 (d) 76 (e) 65 c 2006 Carl James Schwarz 8
22. Using the same data as in the previous question, the 75th percentile is approximately: (a) 76.5 (b) 77.5 (c) 75.0 (d) 78.5 (e) 78.0 23. A sample of 99 distances has a mean of 24 feet and a median of 24.5 feet. Unfortunately, it has just been discovered that an observation which was erroneously recorded as 30 actually had a value of 35. If we make this correction to the data, then: (a) the mean remains the same, but the median is increased (b) the mean and median remain the same (c) the median remains the same, but the mean is increased (d) the mean and median are both increased (e) we do not know how the mean and median are affected without further calculations; but the variance is increased. 24. The term test scores of 15 students enrolled in a Business Statistics class were recorded in ascending order as follows: 4, 7, 7, 9, 10, 11, 13, 15, 15, 15, 17, 17, 19, 19, 20 After calculating the mean, median, and mode, an error is discovered: one of the 15 s is really a 17. The measures of central tendency which will change are: (a) the mean only (b) the mode only (c) the median only (d) the mean and mode (e) all three measures 25. Suppose a frequency distribution is skewed with a median of $75.00 and a mode of $80.00. Which of the following is a possible value for the mean of distribution? (a) $86 (b) $91 (c) $64 c 2006 Carl James Schwarz 9
(d) $75 (e) None of these 26. Earthquake intensities are measured using a device called a seismograph which is designed to be most sensitive for earthquakes with intensities between 4.0 and 9.0 on the open-ended Richter scale. Measurements of nine earthquakes gave the following readings: 4.5 L 5.5 H 8.7 8.9 6.0 H 5.2 where L indicates that the earthquake had an intensity below 4.0 and a H indicates that the earthquake had an intensity above 9.0. The median earthquake intensity of the sample is: (a) Cannot be computed because all of the values are not known (b) 8.70 (c) 5.75 (d) 6.00 (e) 6.47 27. Earthquake intensities are measured using a device called a seismograph which is designed to be most sensitive for earthquakes with intensities between 4.0 and 9.0 on the open-ended Richter scale Measurements of ten earthquakes gave the following readings: 4.5 L 5.5 H 8.7 8.9 6.0 H 5.2 7.2 where L indicates that the earthquake had an intensity below 4.0 and a H indicates that the earthquake had an intensity above 9.0. One measure of central tendancy is the x% trimmed mean computed after trimming x% of the upper values and x% of the bottom values. The value of the 20% trimmed mean is: (a) Cannot be computed because all of the values are not known (b) 6.00 (c) 6.60 (d) 6.92 (e) 6.57 28. When testing water for chemical impurities, results are often reported as bdl, i.e., below detection limit. The following are the measurements of the amount of lead in a series of water samples taken from inner city households (ppm). c 2006 Carl James Schwarz 10
5, 7, 12, bdl, 10, 8, bdl, 20, 6. Which of the following is correct? (a) The mean lead level in the water is about 10 ppm. (b) The mean lead level in the water is about 8 ppm. (c) The median lead level in the water is 7 ppm. (d) The median lead level in the water is 8 ppm. (e) Neither the mean nor the median can be computed because some values are unknown. 29. A clothing and textiles student is trying to assess the effect of a jacket s design on the time it takes preschool children to put the jacket on. In a pretest, she timed 7 children as they put on her prototype jacket. The times (in seconds) are provided below. n n 65 43 n 119 39 The n s represent children who had not put the jacket on after 120 seconds (in which case the children were allowed to stop). Which of the following would be the best value to use as the typical time required to put on the jacket? (a) The median time, which was 43 seconds. (b) The mean time, which was 66 seconds. (c) The median time, which was 52 seconds. ok (d) The median time, which was 119 seconds. ok (e) The missing times (the n s) mean we can t calculate any useful measures of central tendency. 30. For the following histogram, what is the proper ordering of the mean, median, and mode? Note that the graph is NOT numerically precise - only the relative positions are important. c 2006 Carl James Schwarz 11
(a) I = mean II = median III = mode (b) I =mode II = median III = mean (c) I = median II = mean III = mode (d) I = mode II = mean III = median (e) I = mean II = mode III = median 31. The following statistics were collected on two groups of cattle Group A Group B sample size 45 30 sample mean 1000 lbs 800 lbs sample std. dev 80 lbs 70 lbs Which of the following statements is correct? (a) Group A is less variable than Group B because Group A s standard deviation is larger. (b) Group A is relatively less variable than Group B because Group A s coefficient of variation (the ratio of the standard deviation to the mean) is smaller (c) Group A is less variable than Group B because the std deviation per animal is smaller. (d) Group A is relatively more variable than Group B because the sample mean is larger. (e) Group A is more variable than Group B because the sample size is larger. 32. Normal body temperature varies by time of day. A series of readings was taken of the body temperature of a subject. The mean reading was found to be 36.5řC with a standard deviation of 0.3řC. When converted to řf, the mean and standard deviation are: (řf = řc(1.8) + 32). (a) 97.7, 32 (b) 97.7, 0.30 (c) 97.7, 0.54 (d) 97.7, 0.97 (e) 97.7, 1.80 33. A scientist is weighing each of 30 fish. She obtains a mean of 30 g and a standard deviation of 2 g. After completing the weighing, she finds that the scale was misaligned, and always under reported every weight by 2 g, i.e. a fish that really weighed 26 g was reported to weigh 24 g. What is mean and standard deviation after correcting for the error in the scale? [Hint: recall that the mean measures central tendency and the standard deviation measures spread.] c 2006 Carl James Schwarz 12
(a) 28 g, 2 g (b) 30 g, 4 g (c) 32 g, 2 g (d) 32 g, 4 g (e) 28 g, 4 g 34. A researcher wishes to calculate the average height of patients suffering from a particular disease. From patient records, the mean was computed as 156 cm, and standard deviation as 5 cm. Further investigation reveals that the scale was misaligned, and that all reading are 2 cm too large, e.g., a patient whose height is really 180 cm was measured as 182 cm. Furthermore, the researcher would like to work with statistics based on metres. The correct mean and standard deviation are: (a) 1.56m,.05m (b) 1.54m,.05m (c) 1.56m,.03m (d) 1.58m,.05m (e) 1.58m,.07m 35. Rainwater was collected in water collectors at thirty different sites near an industrial basin and the amount of acidity (ph level) was measured. The mean and standard deviation of the values are 4.60 and 1.10 respectively. When the ph meter was recalibrated back at the laboratory, it was found to be in error. The error can be corrected by adding 0.1 ph units to all of the values and then multiply the result by 1.2. The mean and standard deviation of the corrected ph measurements are: (a) 5.64, 1,44 (b) 5.64, 1.32 (c) 5.40, 1.44 (d) 5.40, 1.32 (e) 5.64, 1.20 36. Which of the following statements is NOT true? (a) In a symmetric distribution, the mean and the median are equal. (b) The first quartile is equal to the twenty-fifth percentile. (c) In a symmetric distribution, the median is halfway between the first and the third quartiles. (d) The median is always greater than the mean. (e) The range is the difference between the largest and the smallest observations in the data set. c 2006 Carl James Schwarz 13
37. An experiment was conducted where a person s heart rate was measured 4 times in the space of 10 minutes. This was repeated on a sample of 20 people. Which of the following is not correct? (a) The standard deviation within subjects refers to the repeated measurements of a single person s heart rate. (b) The standard deviation among subjects refers to the variation in heart rates among different people. (c) The variation among subjects was larger than the variation within subjects. (d) The variation in heart rates based on measurements taken for 30 seconds was larger than the variation of heart rates based on measurements taken for 15 seconds. (e) The average of the heart rate computed from the 15 seconds measuring period was about the same as the average of the heart rates computed from the 30 second measurement periods. 38. Here is a summary graph of complex carbohyrates for each of the three fibre groups in the cereal dataset. Which of the following is NOT correct? (a) The low fibre group is more variable than the medium fibre group because the central box is larger. c 2006 Carl James Schwarz 14
(b) About 25% of low fibre cereals have less than 12 g of complex carbohydrates per serving. (c) About 50% of medium fibre cereals have more than 15 g of complex carbohydrates per serving. (d) The average amount of complex carbohydrates per serving for the high fibre group appears to be much smaller than the other two groups. (e) About 25% of the medium fibre cereals have less than 10 g of complex carbohydrates. 39. You are allowed to choose four whole numbers from 1 to 10 (inclusive, without repeats). Which of the following is FALSE? (a) The numbers 4, 5, 6, 7 have the smallest possible standard deviation. (b) The numbers 1, 2, 3, 4 have the smallest possible standard deviation. (c) The numbers 1, 5, 6, 10 have the largest possible standard deviation. (d) The numbers 1, 2, 9, 10 have the largest possible standard deviation. (e) The numbers 7, 8, 9, 10 have the smallest possible standard deviation. 40. Which of the following is FALSE: (a) The numbers 3, 3, 3 have a standard deviation of 0. (b) The numbers 3, 4, 5 have the same standard deviation as 1003, 1004, 1005. (c) The standard deviation is a measure of spread around the centre of the data. (d) The numbers 1, 5, 9 have a smaller standard deviation than 101, 105, 109. (e) The standard deviation can only be computed for interval or ratio scaled data. 41. You are allowed to choose any four integers, without limits but without repeats. Which of the following is FALSE? (a) The numbers 4, 5, 6, 7 has the same standard deviation as the numbers 1231, 1232, 1233, 1234. (b) The numbers 1, 5, 7, 9 has a smaller standard deviation than the numbers 1231, 1235, 1237, 1239. (c) The numbers 1, 5, 6, 10 has a larger standard deviation than the numbers 1231, 1232, 1233, 1234. (d) The numbers 1, 2, 9, 10 has the same standard standard deviation as the numbers 1231, 1232, 1239, 1240. (e) The numbers 1236, 1237, 1238, 1239 has the smallest possible standard deviation. c 2006 Carl James Schwarz 15