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UNIT 3 REVIEW True/False Indicate whether the statement is true or false. ____ 1. The mode is not affected by extreme values in a data set. ____ 2. Because the denominator is n – 1, the sample standard deviation is adjusted to become a greater value than when using the population standard deviation formula. ____ 3. Whenever outliers are present in a set of data, they must be omitted. ____ 4. Standard deviation shows how the data are clustered around the mean. ____ 5. The interquartile range shows how the data are clustered around the median. Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 6. The number of occurrences of a particular value is called the a. range c. tally b. frequency d. data ____ 7. Which question(s) are meant to generate numeric data? I. What is the fuel economy of your car? II. How many cylinders does your car’s engine have? III. Which manufacturer built your car? a. only I c. only III b. only II d. both I and II ____ 8. Statistics is a. the study of bar graphs, histograms, circle graphs and frequency polygons b. the calculation of mean, median and mode c. the gathering, organization, analysis and presentation of numerical information d. a set of data ____ 9. The number of patients treated in a dental office on Mondays was recorded for 11 weeks. What are the mean, median, and mode for this set of data? 5, 17, 28, 28, 28, 15, 13, 18, 10, 16, 20 a. mean 17, median 18, mode 28 b. mean 18, median 17, mode 28 c. mean 16.5, median 18, mode 28 d. mean 28, median 17, mode 18 ____ 10. Which of the following is not a characteristic of the median? a. It is easy to define and easy to understand. b. It is affected by the number of extreme values but not by their values. c. It can be computed when data are grouped. d. It does not exist for some sets of data. Matching Match these formulas to the terms below. a. d. b. e. c. f. ____ 11. population mean, ____ 12. sample mean, ____ 13. population standard deviation, ____ 14. sample standard deviation, s ____ 15. population variance, ____ 16. sample variance, s2 Short Answer 17. What is wrong with the intervals in the following table? Height (cm) 60–62 63–65 66–69 69–75 Frequency 6 18 44 10 18. When recommending five mutual funds for you to consider, a financial planner mentions that the typical minimum investment is $2000 since the minimum investment amounts for the five funds are $500, $2000, $2500, $25 000, and $2000, respectively. However, when you key these numbers into a calculator, you find that the average minimum investment is, in fact, $6400. What accounts for this discrepancy? 19. A building maintenance company tracked the number of months that fluorescent tubes lasted in the different offices in a building. Calculate the mean, median, and mode for this set of data. 8, 29, 22, 15, 10, 22, 12, 4, 22 20. The ages, in years, of a group of friends are listed below. 29, 33, 36, 48, 50, 51, 53, 53 a) Find the mean, median, and mode of the ages. b) Explain what each of these measures tells you about this group of friends. c) What do the relative values of the mean and median tell you about the group? 21. The mean of the values 9, 11, 13, 21, 24, 18, and d is 17. Find d. 22. Each child in a study of infantile autism was given a behavioural test and graded on a scale from 0 (no symptoms) to 116 (maximum severity). The scores of the 21 children in the study were as follows. 27, 35, 65, 67, 47, 46, 63, 44, 34, 51, 17, 40, 41, 60, 24, 48, 29, 73, 60, 41, 47 Calculate the mean, the standard deviation, and the variance. A consumer magazine evaluated 39 models of bathroom scales. The table below lists the prices for each model (rounded to the nearest dollar). Scale Model EconoHealth A10 EconoHealth A12 EconoHealth B10 EconoHealth E10 EconoHealth Digital-10 EconoHealth E-20 EconoHealth E-30 HealthSkale 190 HealthSkale 210 HealthSkale 211 HealthSkale 290 Deluxe HealthSkale 310 HealthSkale 1000 HealthSkale 1002 HydroXact 12573 HydroXact 12756 HydroXact 12856 Prowt P10A Prowt Value Prowt Value 2 Price ($) 50 50 50 28 65 40 50 22 32 30 79 50 23 20 35 24 25 120 35 35 Scale Model Superskale 6400 Superskale 7200 Superskale 8000 Superskale 8280 SvelteChek 12300 SvelteChek 12400D SvelteChek 12509 SvelteChek 12510 SvelteChek Fashion SvelteChek Pro SvelteChek Xtra Weighbeter 550 Weighbeter 801D Weighbeter 830 Weighbeter 835 Weighbeter 950 Weighbeter 2000 Weighbeter 2100 Weighbeter Basic Price ($) 65 20 14 25 24 48 15 10 17 50 25 22 60 30 30 10 12 20 12 23. Find the median, first quartile, and third quartile for the prices of these bathroom scales. 24. Calculate the mean, standard deviation, and variance for the prices of these bathroom scales. 25. What is the z-score for the price of a) the Weighbeter 801D scale? b) the Weighbeter 830 scale? 26. What is the z-score for the price of a) the EconoHealth E10 scale? b) the HydroXact 12573 scale? c) the Prowt P10A scale? 27. Find the range and the interquartile range for the prices of these bathroom scales. Problem 28. The table summarizes data collected in a survey of owners of small trucks. Most owners reported their distances rounded to the nearest 100 km. Distance Travelled Annually (km) 5 000–6 999 7 000–8 999 9 000–10 999 11 000–12 999 13 000 –14 999 15 000–16 999 17 000–18 999 19 000–21 000 Number of Trucks 5 10 12 20 20 14 11 4 Estimate the mean distance these trucks were driven annually. 29. The table lists the approximate numbers of residents in 21 Canadian cities in 2002. City Calgary Edmonton Halifax Hamilton Kingston Kitchener/Waterloo Lethbridge London Ottawa Regina Saint John a) b) c) d) e) f) Population 864 700 693 800 117 200 347 500 60 300 276 400 71 200 350 900 348 500 182 800 73 600 City Saskatoon Sault Sainte Marie St. John's Sudbury Thunder Bay Toronto Vancouver Victoria Windsor Winnipeg Find the median, first quartile, and third quartile for these data. Determine the range and interquartile range. Calculate the mean, standard deviation, and variance. What is the z-score for the population of Windsor? What is the z-score for the population of Toronto? Interpret the z-scores for Toronto and Windsor. Population 72 500 193 600 97 500 99 200 122 500 2 571 700 534 600 76 600 213 100 635 200 UNIT 3 REVIEW Answer Section TRUE/FALSE 1. ANS: OBJ: KEY: 2. ANS: OBJ: KEY: 3. ANS: OBJ: KEY: 4. ANS: OBJ: KEY: 5. ANS: OBJ: KEY: T PTS: 1 Section 2.5 LOC: D1.1 mode | extreme values T PTS: 1 Section 2.6 LOC: D1.1 standard deviation F PTS: 1 Section 2.6 LOC: C1.2 outlier T PTS: 1 Section 2.6 LOC: D1.1 standard deviation | mean T PTS: 1 Section 2.6 LOC: D1.1 interquartile range | median DIF: 1 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 1 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 1 REF: Knowledge & Understanding TOP: Organization of Data for Analysis DIF: 1 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 1 REF: Knowledge & Understanding TOP: Statistical Analysis MULTIPLE CHOICE 6. ANS: OBJ: KEY: 7. ANS: OBJ: KEY: 8. ANS: OBJ: KEY: 9. ANS: OBJ: KEY: 10. ANS: OBJ: KEY: B PTS: 1 Section 2.1 LOC: C1.3 frequency D PTS: 1 Section 2.1 LOC: C1.3 type of data C PTS: 1 Section 2.1 LOC: C1.3 statistics B PTS: 1 Section 2.5 LOC: D1.1 mean | median | mode D PTS: 1 Section 2.5 LOC: D1.1 mean | median | mode DIF: 1 REF: Knowledge & Understanding TOP: Organization of Data for Analysis E PTS: 1 Section 2.6 LOC: D1.1 measures of dispersion F PTS: 1 Section 2.6 LOC: D1.1 measures of dispersion DIF: 2 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 1 REF: Knowledge & Understanding TOP: Organization of Data for Analysis DIF: 1 REF: Knowledge & Understanding TOP: Organization of Data for Analysis DIF: 1 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 2 REF: Knowledge & Understanding TOP: Statistical Analysis MATCHING 11. ANS: OBJ: KEY: 12. ANS: OBJ: KEY: DIF: 2 REF: Knowledge & Understanding TOP: Statistical Analysis 13. ANS: OBJ: KEY: 14. ANS: OBJ: KEY: 15. ANS: OBJ: KEY: 16. ANS: OBJ: KEY: B PTS: 1 Section 2.6 LOC: D1.1 measures of dispersion A PTS: 1 Section 2.6 LOC: D1.1 measures of dispersion D PTS: 1 Section 2.6 LOC: D1.1 measures of dispersion C PTS: 1 Section 2.6 LOC: D1.1 measures of dispersion DIF: 2 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 2 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 2 REF: Knowledge & Understanding TOP: Statistical Analysis DIF: 2 REF: Knowledge & Understanding TOP: Statistical Analysis SHORT ANSWER 17. ANS: The first two intervals should be joined at their endpoints, as should the last two, since height is a continuous variable Also, the intervals have three different widths, so it is not possible to make direct comparisons of the frequencies. PTS: 1 DIF: 2 REF: Knowledge & Understanding OBJ: Section 2.1 LOC: D1.3 TOP: Statistical Analysis KEY: intervals 18. ANS: You and the financial planner have interpreted the word typical differently. The planner was referring to the mode of the minimum investment amounts, while you calculated the mean. PTS: 1 DIF: 3 REF: Application LOC: D1.5 TOP: Statistical Analysis KEY: interpreting statistics | mean | median | mode 19. ANS: mean 16, median 15, mode, 22 OBJ: Section 2.5 PTS: 1 DIF: 1 REF: Knowledge & Understanding OBJ: Section 2.5 LOC: D1.1 TOP: Statistical Analysis KEY: mean | median | mode 20. ANS: a) mean 44.1, median 49, mode 53 b) The mean indicates that the arithmetic average of the ages is about 44. The median indicates that half of the friends are under 49 and the other half are over 49. The mode indicates that 53 is the most common age in the group, but this information is not significant since the group is so small. c) Since the median is higher than the mean, the friends’ ages must be unevenly distributed. PTS: 1 DIF: 3 REF: Application | Communication OBJ: Section 2.5 LOC: D1.1 | D1.5 TOP: Statistical Analysis KEY: interpreting statistics | mean | median | mode 21. ANS: 23 PTS: 1 DIF: 2 REF: Application OBJ: Section 2.5 LOC: D1.5 TOP: Statistical Analysis KEY: mean 22. ANS: Since the study is trying to determine the characteristics of the population of all autistic children, use the formulas for calculating statistics for a sample. = 45.7, s = 15.1, s2 = 229 PTS: 1 DIF: 2 REF: Application LOC: D1.1 TOP: Statistical Analysis 23. ANS: median $30, first quartile $20, third quartile $50 OBJ: Section 2.6 KEY: measures of dispersion PTS: 1 DIF: 2 REF: Application OBJ: Section 2.6 LOC: D1.2 TOP: Statistical Analysis KEY: median | quartiles 24. ANS: The 39 scales in the survey can be considered a sample of the population of all the different bathroom scales on the market. Therefore, use the formulas for calculating statistics for a sample. = $35.18, s = $22.02, s2 = 485 PTS: 1 LOC: D1.1 25. ANS: a) 1.13 b) –0.24 DIF: 2 REF: Application TOP: Statistical Analysis OBJ: Section 2.6 KEY: measures of dispersion PTS: 1 LOC: D1.2 26. ANS: a) –0.33 b) –0.0082 c) 3.85 DIF: 3 REF: Application TOP: Statistical Analysis OBJ: Section 2.6 KEY: z-score PTS: 1 DIF: 3 REF: Application LOC: D1.2 TOP: Statistical Analysis 27. ANS: range $110, interquartile range $30 OBJ: Section 2.6 KEY: z-score PTS: 1 LOC: D1.1 PROBLEM 28. ANS: DIF: 3 REF: Application TOP: Statistical Analysis OBJ: Section 2.6 KEY: measures of dispersion These trucks were driven a mean distance of about 13 000 km annually. PTS: 1 DIF: 2 REF: Application OBJ: Section 2.5 LOC: D1.1 TOP: Statistical Analysis KEY: mean 29. ANS: a) Since there are 21 cities listed, the median is the 11th greatest value in the set of data: 193 600. The first quartile is the midpoint between the fifth and sixth least values, so Q1 = 97 500. Similarly, the third quartile is the midpoint between the fifth and sixth greatest values, so Q3 = 350 900. The median and quartiles can be calculated with a graphing calculator by entering the data into a list and then using the 1-Var Stats function from the STAT CALC menu. In a spreadsheet, you can use the MEDIAN and QUARTILE functions. b) The range is the greatest value minus the least one: 2 571 700 – 60 300 = 2 511 400. The interquartile range is Q3 – Q1 = 253 400. c) The 21 cities can be considered a sample of all the cities in Canada. Therefore, use the sample version of the formulas for the mean, standard deviation, and variance. On a graphing calculator, the 1-VAR Stats function will calculate both and s. In Microsoft® Excel, you can use the AVERAGE, STDEV, and VAR functions to calculate , s, and s2, respectively. In Corel® Quattro® Pro, use @AVG, @STDS, and @VARS. The resulting values are = 381 114, s = 552 863, and s2 = 3.056 575 1011. d) For Windsor, e) For Toronto, f) Windsor’s population is slightly less than the mean population, whereas Toronto’s is significantly greater than the mean. PTS: 1 DIF: 4 REF: Application OBJ: Section 2.6 LOC: D1.1 | D1.2 | D1.5 TOP: Statistical Analysis KEY: measures of dispersion | variance | standard deviation