Descriptive Statistics: Homework 1.9 2008/05/21 15:08:00 GMT-5 2008/08/21 10:04:40.680 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org box descriptive deviation elementary statistics exercise frequency histogram homework mean median mode percentile plot population problem quartile relative sample standard statistics This module provides homework questions related to lessons on descriptive statistics. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows: # of movies Frequency Relative Frequency Cumulative Relative Frequency 0 5 1 9 2 6 3 4 4 1 aFind the sample mean x bFind the sample standard deviation, s cConstruct a histogram of the data. dComplete the columns of the chart. eFind the first quartile. fFind the median. gFind the third quartile. hConstruct a box plot of the data. iWhat percent of the students saw fewer than three movies? jFind the 40th percentile. kFind the 90th percentile. a1.48 b1.12 e1 f1 g2 h i80% j1 k3 The median age for U.S. blacks currently is 30.1 years; for U.S. whites it is 36.6 years. (Source: U.S. Census) aBased upon this information, give two reasons why the black median age could be lower than the white median age. bDoes the lower median age for blacks necessarily mean that blacks die younger than whites? Why or why not? cHow might it be possible for blacks and whites to die at approximately the same age, but for the median age for whites to be higher? Forty randomly selected students were asked the number of pairs of sneakers they owned. Let X = the number of pairs of sneakers owned. The results are as follows: X Frequency Relative Frequency Cumulative Relative Frequency 1 2 2 5 3 8 4 12 5 12 7 1 aFind the sample mean x bFind the sample standard deviation, s cConstruct a histogram of the data. dComplete the columns of the chart. eFind the first quartile. fFind the median. gFind the third quartile. hConstruct a box plot of the data. iWhat percent of the students owned at least five pairs? jFind the 40th percentile. kFind the 90th percentile. a3.78 b1.29 e3 f4 g5 h i32.5% j4 k5 600 adult Americans were asked by telephone poll, What do you think constitutes a middle-class income? The results are below. Also, include left endpoint, but not the right endpoint. (Source: Time magazine; survey by Yankelovich Partners, Inc.) "Not sure" answers were omitted from the results. Salary ($) Relative Frequency < 20,000 0.02 20,000 - 25,000 0.09 25,000 - 30,000 0.19 30,000 - 40,000 0.26 40,000 - 50,000 0.18 50,000 - 75,000 0.17 75,000 - 99,999 0.02 100,000+ 0.01 aWhat percent of the survey answered "not sure" ? bWhat percent think that middle-class is from $25,000 - $50,000 ? cConstruct a histogram of the data iShould all bars have the same width, based on the data? Why or why not? iiHow should the <20,000 and the 100,000+ intervals be handled? Why? dFind the 40th and 80th percentiles Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year (Source: San Jose Mercury News). 177 205 210 210 232 205 185 185 178 210 206 212 184 174 185 242 188 212 215 247 241 223 220 260 245 259 278 270 280 295 275 285 290 272 273 280 285 286 200 215 185 230 250 241 190 260 250 302 265 290 276 228 265aOrganize the data from smallest to largest value. bFind the median. cFind the first quartile. dFind the third quartile. eConstruct a box plot of the data. fThe middle 50% of the weights are from _______ to _______. gIf our population were all professional football players, would the above data be a sample of weights or the population of weights? Why? hIf our population were the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why? iAssume the population was the San Francisco 49ers. Find:ithe population mean, μ . iithe population standard deviation, σ . iiithe weight that is 2 standard deviations below the mean. ivWhen Steve Young, quarterback, played football, he weighed 205 pounds. How many standard deviations above or below the mean was he? jThat same year, the average weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer? b241 c205.5 d272.5 e f205.5, 272.5 gsample hpopulation ii236.34 ii37.50 iii161.34 iv0.84 std. dev. below the mean jYoung An elementary school class ran 1 mile in an average of 11 minutes with a standard deviation of 3 minutes. Rachel, a student in the class, ran 1 mile in 8 minutes. A junior high school class ran 1 mile in an average of 9 minutes, with a standard deviation of 2 minutes. Kenji, a student in the class, ran 1 mile in 8.5 minutes. A high school class ran 1 mile in an average of 7 minutes with a standard deviation of 4 minutes. Nedda, a student in the class, ran 1 mile in 8 minutes. aWhy is Kenji considered a better runner than Nedda, even though Nedda ran faster than he? bWho is the fastest runner with respect to his or her class? Explain why. In a survey of 20 year olds in China, Germany and America, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results. aIn complete sentences, describe what the shape of each box plot implies about the distribution of the data collected. bExplain how it is possible that more Americans than Germans surveyed have been to over eight foreign countries. cCompare the three box plots. What do they imply about the foreign travel of twenty year old residents of the three countries when compared to each other? Twelve teachers attended a seminar on mathematical problem solving. Their attitudes were measured before and after the seminar. A positive number change attitude indicates that a teacher's attitude toward math became more positive. The twelve change scores are as follows: 3 8 -1 2 0 5 -3 1 -1 6 5 -2 aWhat is the average change score? bWhat is the standard deviation for this population? cWhat is the median change score? dFind the change score that is 2.2 standard deviations below the mean. Three students were applying to the same graduate school. They came from schools with different grading systems. Which student had the best G.P.A. when compared to his school? Explain how you determined your answer. Student G.P.A. School Ave. G.P.A. School Standard Deviation Thuy 2.7 3.2 0.8 Vichet 87 75 20 Kamala 8.6 8 0.4 Kamala Given the following box plot: aWhich quarter has the smallest spread of data? What is that spread? bWhich quarter has the largest spread of data? What is that spread? cFind the Inter Quartile Range (IQR). dAre there more data in the interval 5 - 10 or in the interval 10 - 13? How do you know this? eWhich interval has the fewest data in it? How do you know this?I 0-2II2-4III10-12IV12-13 Given the following box plot: aThink of an example (in words) where the data might fit into the above box plot. In 2-5 sentences, write down the example. bWhat does it mean to have the first and second quartiles so close together, while the second to fourth quartiles are far apart? Santa Clara County, CA, has approximately 27,873 Japanese-Americans. Their ages are as follows. (Source: West magazine) Age Group Percent of Community 0-17 18.9 18-24 8.0 25-34 22.8 35-44 15.0 45-54 13.1 55-64 11.9 65+ 10.3 aConstruct a histogram of the Japanese-American community in Santa Clara County, CA. The bars will not be the same width for this example. Why not? bWhat percent of the community is under age 35? cWhich box plot most resembles the information above? Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, each asked adult consumers the number of fiction paperbacks they had purchased the previous month. The results are below. Publisher A # of books Freq. Rel. Freq. 0 10 1 12 2 16 3 12 4 8 5 6 6 2 8 2 Publisher B # of books Freq. Rel. Freq. 0 18 1 24 2 24 3 22 4 15 5 10 7 5 9 1 Publisher C # of books Freq. Rel. Freq. 0-1 20 2-3 35 4-5 12 6-7 2 8-9 1 aFind the relative frequencies for each survey. Write them in the charts. bUsing either a graphing calculator, computer, or by hand, use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of 1. For Publisher C, make bar widths of 2. cIn complete sentences, give two reasons why the graphs for Publishers A and B are not identical. dWould you have expected the graph for Publisher C to look like the other two graphs? Why or why not? eMake new histograms for Publisher A and Publisher B. This time, make bar widths of 2. fNow, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer. Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all on-board transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Below is a summary of the bills for each group. Singles Amount($) Frequency Rel. Frequency 51-100 5 101-150 10 151-200 15 201-250 15 251-300 10 301-350 5 Couples Amount($) Frequency Rel. Frequency 100-150 5 201-250 5 251-300 5 301-350 5 351-400 10 401-450 10 451-500 10 501-550 10 551-600 5 601-650 5 aFill in the relative frequency for each group. bConstruct a histogram for the Singles group. Scale the x-axis by $50. widths. Use relative frequency on the y-axis. cConstruct a histogram for the Couples group. Scale the x-axis by $50. Use relative frequency on the y-axis. dCompare the two graphs:iList two similarities between the graphs. iiList two differences between the graphs. iiiOverall, are the graphs more similar or different? eConstruct a new graph for the Couples by hand. Since each couple is paying for two individuals, instead of scaling the x-axis by $50, scale it by $100. Use relative frequency on the y-axis. fCompare the graph for the Singles with the new graph for the Couples:iList two similarities between the graphs. iiOverall, are the graphs more similar or different? iBy scaling the Couples graph differently, how did it change the way you compared it to the Singles? jBased on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person in a couple? Explain why in one or two complete sentences. Refer to the following histograms and box plot. Determine which of the following are true and which are false. Explain your solution to each part in complete sentences. aThe medians for all three graphs are the same. bWe cannot determine if any of the means for the three graphs is different. cThe standard deviation for (b) is larger than the standard deviation for (a). dWe cannot determine if any of the third quartiles for the three graphs is different. aTrue bTrue cTrue dFalse Refer to the following box plots. aIn complete sentences, explain why each statement is false. iData 1 has more data values above 2 than Data 2 has above 2. iiThe data sets cannot have the same mode. iiiFor Data 1, there are more data values below 4 than there are above 4. bFor which group, Data 1 or Data 2, is the value of “7” more likely to be an outlier? Explain why in complete sentences In a recent issue of the IEEE Spectrum, 84 engineering conferences were announced. Four conferences lasted two days. Thirty-six lasted three days. Eighteen lasted four days. Nineteen lasted five days. Four lasted six days. One lasted seven days. One lasted eight days. One lasted nine days. Let X = the length (in days) of an engineering conference. aOrganize the data in a chart. bFind the median, the first quartile, and the third quartile. cFind the 65th percentile. dFind the 10th percentile. eConstruct a box plot of the data. fThe middle 50% of the conferences last from _______ days to _______ days. gCalculate the sample mean of days of engineering conferences. hCalculate the sample standard deviation of days of engineering conferences. iFind the mode. jIf you were planning an engineering conference, which would you choose as the length of the conference: mean; median; or mode? Explain why you made that choice. kGive two reasons why you think that 3 - 5 days seem to be popular lengths of engineering conferences. b4,3,5 c4 d3 e f3,5 g3.94 h1.28 i3 jmode A survey of enrollment at 35 community colleges across the United States yielded the following figures (source: Microsoft Bookshelf): 6414 1550 2109 9350 21828 4300 5944 5722 2825 2044 5481 5200 5853 2750 10012 6357 27000 9414 7681 3200 17500 9200 7380 18314 6557 13713 17768 7493 2771 2861 1263 7285 28165 5080 11622 aOrganize the data into a chart with five intervals of equal width. Label the two columns "Enrollment" and "Frequency." bConstruct a histogram of the data. cIf you were to build a new community college, which piece of information would be more valuable: the mode or the average size? dCalculate the sample average. eCalculate the sample standard deviation. fA school with an enrollment of 8000 would be how many standard deviations away from the mean? The median age of the U.S. population in 1980 was 30.0 years. In 1991, the median age was 33.1 years. (Source: Bureau of the Census) aWhat does it mean for the median age to rise? bGive two reasons why the median age could rise. cFor the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not? cMaybe A survey was conducted of 130 purchasers of new BMW 3 series cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of new BMW 7 series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results. aIn complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series. bWhich group is most likely to have an outlier? Explain how you determined that. cCompare the three box plots. What do they imply about the age of purchasing a BMW from the series when compared to each other? dLook at the BMW 5 series. Which quarter has the smallest spread of data? What is that spread? eLook at the BMW 5 series. Which quarter has the largest spread of data? What is that spread? fLook at the BMW 5 series. Find the Inter Quartile Range (IQR). gLook at the BMW 5 series. Are there more data in the interval 31-38 or in the interval 45-55? How do you know this? hLook at the BMW 5 series. Which interval has the fewest data in it? How do you know this? i31-35ii38-41iii41-64 The following box plot shows the U.S. population for 1990, the latest available year. (Source: Bureau of the Census, 1990 Census) aAre there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know? b12.6% are age 65 and over. Approximately what percent of the population are of working age adults (above age 17 to age 65)? amore children b62.4% Javier and Ercilia are supervisors at a shopping mall. Each was given the task of estimating the mean distance that shoppers live from the mall. They each randomly surveyed 100 shoppers. The samples yielded the following information: Javier Ercilla x 6.0 miles 6.0 miles s 4.0 miles 7.0 miles aHow can you determine which survey was correct ? bExplain what the difference in the results of the surveys implies about the data. cIf the two histograms depict the distribution of values for each supervisor, which one depicts Ercilia's sample? How do you know?

dIf the two box plots depict the distribution of values for each supervisor, which one depicts Ercilia’s sample? How do you know?

Student grades on a chemistry exam were: 77, 78, 76, 81, 86, 51, 79, 82, 84, 99aConstruct a stem-and-leaf plot of the data. bAre there any potential outliers? If so, which scores are they? Why do you consider them outliers? b51,99

Try these multiple choice questions.The next three questions refer to the following information. 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 section. Number of years Frequency Total = 20 7 1 14 3 15 1 18 1 19 4 20 3 22 1 23 1 26 1 40 2 42 2 What is the IQR? A8 B11 C15 D35 A What is the mode? A19 B19.5 C14 and 20 D22.65 A Is this a sample or the entire population? Asample Bentire population Cneither B The next two questions refer to the following table. X = the number of days per week that 100 clients use a particular exercise facility. X Frequency 0 3 1 12 2 33 3 28 4 11 5 9 6 4 The 80th percentile is: A5 B80 C3 D4 D The number that is 1.5 standard deviations BELOW the mean is approximately: A0.7 B4.8 C-2.8 DCannot be determined A The next two questions refer to the following histogram. Suppose one hundred eleven people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each. The percent of people that own at most three (3) T-shirts costing more than $19 each is approximately: A21 B59 C41 DCannot be determined C If the data were collected by asking the first 111 people who entered the store, then the type of sampling is: Acluster Bsimple random Cstratified Dconvenience D