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Elementary Statistics Practice Test 1 Chapters 1 and 2 Chapter 1 1. A marketing company is interested in the proportion of people who will buy a particular product. Identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the possible data values, and f. the data type. Solution: a. all people (maybe in a certain geographic area, such as the United States); b. the group of the people sampled by the marketing company; c. the proportion of all people in the population who will buy the product; d. the proportion of the sample who will buy the product; e. buy, not buy; f. categorical 2. Identify the data type: quantitative discrete, quantitative continuous, or categorical. (a) number of students enrolled at a college (b) brand of toothpaste (c) percent of body fat Solution: a. quantitative discrete; b. categorical; c. quantitative continuous (this calculation involves measurements) 3. A reporter collects a sample of students at a local college by randomly selecting a proportional number of students from each major, including “undecided”. Identify the sampling method. Solution: stratefied sampling (the research may need to take care with students with double-majors, since they have a higher chance of being selected) 4. Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI’s Uniform Crime Report. One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates. Which of the potential problems with samples discussed in Section 1.2 could explain this connection? Elementary Statistics Practice Test 1 - Page 2 Chapters 1 and 2 Solution: Correlation versus causality: The fact that two variables are related does not guarantee that one variable is influencing the other. We cannot assume that crime rate impacts education level or that education level impacts crime rate. Confounding: There are many factors that define a community other than education level and crime rate. Communities with high crime rates and high education levels may have other lurking variables that distinguish them from communities with lower crime rates and lower education levels. Because we cannot isolate these variables of interest, we cannot draw valid conclusions about the connection between education and crime. Possible lurking variables include police expenditures, unemployment levels, region, average age, and size. 5. Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown below. Number of flosses per week 0 1 3 6 7 Frequency 27 18 Relative Frequency 0.45 Cumulative Relative Frequency 0.933 3 1 0.05 0.017 (a) Complete the table. (b) What percent of adults flossed at least six times per week? (c) What percent of adults flossed at most three times per week? Solution: Flosses per wk Freq. 0 27 1 18 a. 3 11 6 3 7 1 Rel. Freq. 0.45 0.3 0.183 0.05 0.017 Cum. Rel. Freq. 0.45 0.75 0.933 0.983 1 b. 6.7%; c. 93.3% 6. How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation. Identify explanatory and response variables, treatments, experimental units, and the control (placebo) group. Elementary Statistics Practice Test 1 - Page 3 Chapters 1 and 2 Solution: Explanatory variable: amount of sleep Response variable: performance measured in assigned tasks Treatments: normal sleep and 27 hours of total sleep deprivation Experimental Units: 19 professional drivers Control/Placebo: completing the experimental session under normal sleep conditions 7. The graph in below shows the number of complaints for six different airlines as reported to the US Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the most complaints? Solution: You cannot assume that the numbers of complaints reflect the quality of the airlines. The airlines shown with the greatest number of complaints are the ones with the most passengers. You must consider the appropriateness of methods for presenting data; in this case displaying totals is misleading. Chapter 2 8. Create a stemplot for the miles per gallon rating for 30 cars as shown below (lowest to highest). 19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43. Stem 1 2 Solution: 3 4 Leaf 999 0115556689 11223456778888 133 Elementary Statistics Practice Test 1 - Page 4 Chapters 1 and 2 9. Examine the overlay of relative frequency polygons in Example 2.11 (Figure 2.9). The Final Test Grades are represented by the dark blue line. The Final Grades are represented by the light blue line. a. Judging by the frequency polygons, which do you think is greater: the mean of the final test grades or the mean of the final grades? b. Use class midpoints and the frequencies in Table 2.16 to estimate the mean of the final test grades. c. Use class midpoints and the frequencies in Table 2.17 to estimate the mean of the final grades. d. Do your estimates agree with your conclusion in part (a)? e. Why are the means found in parts (b) and (c) estimates and not exact mean scores? Solution: a. It appears that the final test grades have a higher mean. At the upper end, there is a class (centered at 94.5) that has than 10 more test grades than final grades. At the lower end, there is a class (centered at 54.5) that has 5 more final grades than test grades. b. 79.5 (Use the midpoints of each class as the “x” value, then find the mean of the frequency table as usual.) c. 77 d. Yes e. We don’t have the actual list of test grades, so we can’t find the exact mean. We are using the midpoint of each class to approximate every grade that falls in that class. For example, we are using 74.5 to approximate all 30 final test grades between 69.5 and 79.5, whereas the actual grades are most likely not all 74.5. 10. We are interested in the number of years students in a particular elementary statistics class have lived in California. The information in the following table is from the entire class. Number of years Freq. Number of Years Freq. 7 1 22 1 14 3 23 1 15 1 26 1 18 1 40 2 19 4 42 2 20 3 Total = 20 Elementary Statistics Practice Test 1 - Page 5 Chapters 1 and 2 a. List the data values sorted smallest to largest. b. Use the data to make a boxplot on your calculator. List the 5 number summary, the 5 values marked on the boxplot, using appropriate symbols. c. 75% of students in this class have lived in California longer than years. d. Find and interpret P80 . e. Find and interpret the percentile of 20 years. f. Is this population or sample data? g. Find the mean and standard deviation of the data. Use the appropriate symbols and units. h. Find the z scores of 7 years and 26 years. Use two decimal places. Which is more unusual relative to the rest of the class? i. Find the number of years living in California that is two standard deviations below the mean. j. Make a modified boxplot of the data on your calculator. (This is the fourth graphing option; it looks like a boxplot, but has extra dots. These dots represent what the calculator decides are outliers.) (a) Which data values does the calculator list as outliers? (Use “TRACE”.) (b) Does the formula the book suggests for identifying outliers (using the IQR) agree with this choice of outliers? (Show your work.) k. Which measure of center is the most representative of this data, the mean or the median? Explain. Solution: a. 7, 14, 14, 14, 15, 18, 19, 19, 19, 19, 20, 20, 20, 22, 23, 26, 40, 40, 42, 42 b. min = 7, Q1 = 16.5, med = 19.5, Q3 = 24.5, max = 42 c. 16.5 d. P80 = 33 years. 80% of students in this class have lived in California shorter than 33 years. 20% of students in this class have lived in California longer than 33 years. e. 20 years marks the 58th percentile, or P58 = 20 years. 58% of students in this class have lived in California less than 20 years; 42% for longer than 20 years. f. This is population data. (The description says that we are interested in this particular class only, and this is all the data from this class.) Elementary Statistics Practice Test 1 - Page 6 Chapters 1 and 2 g. µ = 22.7years, σ = 10.0years. (Note that we are using the population parameter symbols and the population formula for standard deviation from the calculator. Round to one decimal place, since the data has zero decimal places.) h. 7 years: z = −1.57. 26 years: z = 0.34. 7 years is more unusual relative to the rest of the class because it is more standard deviations from the class mean. (Remember not to round until the end of the calculation.) i. 2.7 years is 2 standard deviations below the mean. j. (a) 40, 42 (b) Yes. According to the formula, low outliers are below 16.5 − 1.5(8) = 4.5 years and high outlliers are above 24.5 + 1.5(8) = 36.5 years. k. The median is the best representative of the bulk of the data because the outliers skew the mean to the right, making it appear that the students have lived in California longer than they actually have. Only 6 students have lived in California longer than the mean of 22.7 years. Many of these problems are from Barbara Illowsky & Susan Dean. “Introductory Statistics.” OpenStax College, 2013. iBooks. (Chapters 1 and 2) https://itun.es/us/kFeL1.l