2.6.2: Exercise 2.6

MAKING CONNECTIONS TO THE COLLABORATION

(1) Which of the following was one of the main mathematical ideas of the collaboration?

(i) Any of the three measures of central tendency (mean, median, and mode) are good representations of data. It does not matter which one you use.

(ii) U.S. college students carry far too much credit card debt.

(iii) The mean is calculated by adding all the numbers and dividing by the number of data points.

(iv) The mean, median, and mode all give important information about a data set, but they do not give a complete picture of the data set.

DEVELOPING SKILLS AND UNDERSTANDING

(2) Use the following data set to answer the questions.

13 15 20 20 20 20 20 20 20 23 27 31

(a) What is the mean? Round to two decimal places.

(b) What is the mode?

(c) What is the median?

(d) What fraction of the numbers in the data set is less than the median?

(e) What fraction of the numbers in the data set is greater than the median?

(f) Which of the following statements is correct?

(i) The median is the middle of a data set. Half of the data points are always less than the median, and half are always greater than the median.

(ii) The median is the middle of a data set. When listed in increasing order, there are the same number of data points to the left and to the right of the median..

(iii) The median is the middle of a data set. At least half of the data points are always equal to the median.

(iv) The median is not the middle of a data set. You cannot predict the distribution of the numbers in relationship to the median.

(3) Consider the statement “In the U.S. in 2021, there were $4.1 trillion in credit card transactions.”

(a) What was the daily average dollar amount of transactions in 2021? Round to the nearest hundred million dollars.

(b) How many dollars were spent in credit card transactions each day?

(i) $11,232,876

(ii) $1,460,000,000

(iii) $6,849,315,068

(iv) It is impossible to know.

(4) Students at Dover Community College have a mean credit card debt of $3,600 with a median of $1,500. Students at Ralton Community College have a mean credit card debt of $3,000 with a median of $2,800. Which of the following statements are true based on this information? There may be more than one correct answer.

(i) Less than half of Dover students have debt more than $3,600.

(ii) No more than half of Ralton students have debt less than $2,800.

(iii) About three-fourths of Dover students have debt less than $3,600.

(iv) The largest debt of the Ralton students is less than the largest debt of the Dover students.

(v) The total debt of Ralton students is less than the total debt of Dover students.

(5) Decide whether the following statements are true or not, based on the information provided. Be prepared to explain your reasoning.

(a) The median of 25 numbers is 55. Twelve of the numbers must be greater than 55.

(i) True

(ii) False

(b) The average of 11 numbers is 130. None of the 11 numbers are more than 260.

(i) True

(ii) False

(c) The average of 25 numbers is 100, and the median of those 25 numbers is also 100. The mode of the 25 numbers must be 100.

(i) True

(ii) False

(d) The mean of 45 numbers is 70. If you pick any group of 10 numbers from the 45, the mean will be 70.

(i) True

(ii) False

(e) The average of 42 numbers is 20. The sum of all 42 numbers is 840.

(i) True

(ii) False

(f) The average of 49 numbers is 100. If a 50th number is added and the average remains at 100, the 50th number must have been 100.

(i) True

(ii) False

(6) Rio Blanca City Hall publishes the following statistics on household incomes of the town’s citizens. (The mode is given as a range.)

Mean: $257,000 Median: $65,000 Mode: $20,000–$30,000

Which measure would be the most useful for each of the following situations?

(a) State officials want to estimate the total amount of state income tax paid by the citizens of Rio Blanca.

(i) Mean

(ii) Median

(iii) Mode

(b) The school district wants to know the income level of the largest number of students.

(i) Mean

(ii) Median

(iii) Mode

(c) A businesswoman is thinking about opening an expensive restaurant in the town. She wants to know how many people in town could afford to eat at her restaurant.

(i) Mean

(ii) Median

(iii) Mode

MAKING CONNECTIONS ACROSS THE COURSE

(7) Use Figures 1 and 2 for the following questions.

Figure 1

Figure 2

(a) Select the figure(s) with 40% shaded.

(i) Figure 1

(ii) Figure 2

(iii) Both Figure 1 and Figure 2

(iv) Neither Figure 1 nor Figure 2

(b) Which of the following statements are correct? There may be more than one correct answer.

(i) The shaded area of Figure 1 is larger than the shaded area of Figure 2 because Figure 1 is larger than Figure 2.

(ii) The shaded area of Figure 1 is the same as the shaded area of Figure 2 because they are both 40% of the square.

(iii) The shaded area of Figure 1 is the same proportion of the figure as the shaded area of Figure 2 because they are both 40% of the square.

(c) These figures illustrate what important concept?

(i) Percentages cannot be used for comparisons unless the reference values are equal.

(ii) Percentages compare measures relative to the size of the reference values, but do not give information about absolute measures.

(iii) Percentages are a ratio out of 100, so they can always be compared directly. In other words, 60% of one value is equal to 60% of another value.

(8) Animal population densities are important to those who manage wildlife. This is particularly true when animals pose risks to humans. Managing the population of grizzly bears in Yellowstone National Park is one example. Grizzly bears roam Yellowstone Park, which is 3,472 square miles. Additionally, the bears roam a surrounding area that is 40% as large as Yellowstone. Although their territories overlap considerably, each bear needs at least 10 square miles to himself/herself to call home.¹⁹

(a) Calculate the total available land in and around Yellowstone for grizzly bears to roam. Round to the nearest square mile.

(b) Estimate the maximum number of bears that can be supported on the available land.

(c) If there are about 3 adult female bears for every 2 adult male bears, then approximately how many male bears can be supported in and around Yellowstone Park?

(d) How many female bears?