QR Review Sheet for Final Exam, Spring 2005

1. The following is a list of commuting times (in minutes) for workers in downtown Boston:

35 20 25 40 60 30 50 75 60 50

a) Calculate the mean and median for this data.

b) Use the data to generate a histogram.

c) Construct a 60-second summary, describing any patterns in the data.

2. The following frequency table shows the ages of students in a Quantitative Reasoning class.

Age interval	frequency
18 – 20	2
21 – 23	4
24 – 26	1
27 - 29	3
30 – 32	0
33 – 35	2

a) Find relative frequencies.

b) Use Excel to make a relative frequency histogram of this data.

c) Estimate the mean age for this class.

3. Two histograms derived from US Census data are shown below. This random sample includes 1000 people of age 15 years and older.

Distribution of Ages of Men Distribution of Ages of Women

Use the histograms to approximate the following:

a) the number of men between the ages of 55 and 60.

b) the percentage of women between the ages of 30 and 40.

c) the total number of people aged 60 and over.

4. The following graph is taken from a special report by the Census Bureau titled “Demographic Trends in the 20^th Century”. The graph shows the population density from 1950 to 2000 by metropolitan area.

Use the graph to write an analysis of changes in population density in the United States during this time periods. Make specific references to quantitative data that you see in the graph

5. You are a newspaper reporter writing an article on the average income of people in the Boston area. Which “average” (mean or median) would you use in your article? Explain your reasoning carefully.

6. a) State the definition of a function.

b) Determine in each case whether Q is a function of t. If not, state a reason.

i)

ii). The chart below gives the height (H) and weight (W) for students enrolled in

a seminar. Is weight a function of height? Explain your answer with reference to the data.

H	68 in.	70 in.	67 in.	71 in.	64 in.	70 in.
W	160 lbs.	140 lbs.	130 lbs.	155 lbs.	105 lbs.	145 lbs.

iv) Q represents the amount of gas in the tank of a car over the period of a year.

7. Open the data file MENSMILE.xls. This data file shows the record times for the men’s mile.

a) Insert a new column, titled “years since 1913”. Use Excel to calculate the years since 1913 in that column.

b) Make a scatter plot of the men’s mile record versus years since 1913. Label your graph clearly and make sure your name is on the graph.

c) Assume that the data are linear and fit a linear least squares regression line to the data. Using the variable m for record time and t for years since 1913, record the regression equation and correlation coefficient for the data in the space below (Round off numbers in the equation to 3 decimal places.).

equation: ____________________ correlation coefficient = ______

d) Using a word description interpret the slope and explain what the correlation coefficient tells you about the regression line associated with the data.

e) Using the regression equation, answer the following questions:

i) In what year would your linear model predict a winning time of 3.00 minutes?

ii) What does your linear model predict as the record time for the year 1965?

8. The following table shows the number of cigarettes consumed in the U.S., along with the Census Bureau’s estimate of the U.S. population at that time.

Year	U.S. Consumption of Cigarettes	U.S. Population
1960	484,400,000,000	180,000,000
1970	536,400,000,000	204,000,000
1980	631,500,000,000	227,200,000
1990	525,000,000,000	249,400,000
1997	480,000,000,000	267,800,000

a) What was the average rate of change in cigarette consumption between 1960 and 1980? Between 1960 and 1997? Between 1980 and 1997?

b) Create a 4^th column in this table and calculate the number of cigarette smoker per person for each of the years. What are the units here?

c) Calculate the rate of change in number of cigarettes smoked per person between 1960 and 1997.

c) The total number of cigarettes consumed in the U.S. in 1960 is very close to the number of cigarettes consumed in 1997. Does that mean that smoking was as popular in 1997 as it was in 1960? Explain your answer, making specific references to the data and to your calculations.

9. In 1997 AT&T offered two long distance calling plans. The “One Rate” plan charged a flat rate of $0.15 per minute. The “One Rate Plus” plan charged a service fee of $4.95 a month plus $0.10 per minute. Consider the monthly telephone cost as a function of minutes for both plans and construct a cost function for both plans. Use C₁ to represent the monthly cost of the “One Rate” plan and C₂ to represent the monthly cost of the “One Rate Plus” plan and t to represent the time, in minutes. Find the linear functions for C₁ and C_2.

10. Compute the indicated operations and express the result in scientific notation.

a) (3.54 x 10²¹)(8.9 x 10^-66)

b) (7.9 x 10⁸⁹) / (1.33 x 10⁵⁵)

c) (6.7 x 10 ⁷³) ³³

11. Computer company Alpha is offering the following deal: For only $399, you can buy a brand-new computer. However, you must also buy their internet service, which costs $29.99 per month.

a) Write an equation that describes the amount of money you pay based on the number of months of internet service that you buy.

b) Identify each of the following in your equation, including units.

Independent variable: ____________ Dependent variable: _____________

Rate of change (slope): ___________ Vertical intercept: ___________

c) The fine print says that you must buy at least 24 months of internet service. With this in mind, what is a reasonable domain for your equation from part a? You may also want to consider the lifetime of a computer. You must explain your answer.

d) Computer company Beta offers the same computer for lease for $49.99 per month.

i) Write an equation that describes the amount of money you pay based on the number of months you lease the computer from Beta.

ii) Use Excel to create a spreadsheet showing the total amount of money paid under each plan for 24 months. Make a scatter plot of the data and insert it into your worksheet. Label your graph carefully. Put your name on top of the worksheet and use print preview to make sure that it will print properly.

e) Which is the most cost effective plan? Explain your answer with reference to the equations and graph you made.

12. A car is traveling at a rate of 58 miles per hour. Express the speed of the car in kilometers per minute.

13. The earth has an approximate diameter of 2 * 10⁷ meters. A hydrogen atom has a diameter of approximately .0000000000529 meters. How many objects the size of a hydrogen atom would you need to create a line as long as the earth’s diameter? Write your answer in scientific notation.

14. The sun is approximately 3 orders of magnitude larger in diameter than the earth. The diameter of the earth is 6,300,000 meters. Approximately how large is the sun's diameter?

15. The CEO of a major corporation has a total compensation package (salary plus stocks) of $785,000,000 per year. A worker in a factory owned by the corporation has a total compensation package of $28,000 per year. How many orders of magnitude are there between these two incomes? Write your answer in a complete sentence.

16. The projected budget for the United States in fiscal year 2005 is 2.4 trillion dollars. Approximately 13% of this budget will be used to pay interest on the federal debt. Estimate the amount of money this represents.

17. Last year, the Massachusetts state budget had a 1.4 billion dollar shortfall. That is, the expenditures of the state were $1.4 billion dollars more than the state’s revenues. The state’s population was about 6.4 million people. If the shortfall were spread out evenly among the state’s population, how much would it be per person?

18. China is the most populous country in the world. In 1998 it had about 1.237 billion people. By the end of 1999 the population had grown to 1.267 billion. One could describe this change as either an increase of 0.030 billion (= 30 million) people, or as an increase of 2.4%. Use this information to construct models predicting the size of China’s population in the future.

a) Identify your variables and units.

b) Construct a linear model.

c) Construct an exponential model.

d) Use Excel to construct a spreadsheet showing growth under both models. Create three columns, one labeled “years since 1998”, one labeled “linear growth”, and one labeled “exponential growth”. In the years column, use Excel to fill in years 0 to 20. Then have Excel use your functions to fill in the values for the other two columns.

e) Make one scatter plot displaying both graphs. Label your graphs and axes as usual, and make sure your name is on the graph and the spreadsheet.

d) (Show all work here) What will China’s population be in the year 2050 according to

i) the linear model

ii) the exponential model

19. The half-life of a radioactive isotope is 30 years.

a) Explain what that means.

b) If the initial population is 180 grams, what is the population after 30 years? After 60 years? After 90 years?

c) Write an exponential equation giving the amount of the substance after t years.

20. Open the Excel data file WORLDPOP.xls. This data set shows the world population in millions since 1800. Let t = years since 1880 and make a new column showing t. Then make a scatter plot of t and the population P.

a) Have Excel generate a best fit exponential function. What is the equation

that Excel finds?

b) What is the domain for this equation? The range?

c) What does your model give you for the growth factor?

d) What is the percentage growth rate?

e) Use your model to predict the population in the year 2050.

21. The following is taken from an article that appeared in the New York Times on Tuesday, May 21, 2002. Read the excerpt, and use it to answer the questions below.

The number of people on welfare declined slightly in the final months of 2001 despite the sluggish economy and job losses caused by the Sept. 11 attacks, the Bush administration said today.

But the decline was much more modest than in prior years, suggesting that the number of recipients may have leveled off after declining precipitously since 1994, two years before approval of the landmark 1996 welfare law.

From September to December of last year, the number of welfare recipients declined by 52,006, or 1 percent, to 5,284,711, the Department of Health and Human Services said.

About 12.2 million people were on welfare in August 1996, when President Bill Clinton signed the welfare bill championed by Republicans in Congress. Since then, the number of welfare recipients has dropped by nearly 7 million, or 57 percent — much more than many experts had expected.

The House passed legislation last week to increase work requirements for welfare recipients, thus keeping up the pressure on states to move people from welfare to jobs.

The national unemployment rate rose significantly in the final months of 2001, to 5.8 percent in December, from 5 percent in September. The fact that the federal welfare rolls did not increase suggests that a slowdown in the economy may have less effect on the welfare rolls than many people believed, officials said.

a) How many people were on welfare in August, 1996?

b) Calculate the average annual rate of change in the number of people on welfare between 1996 and 2002.

c) Use your answers from a) and b) to find a linear function that models the number of people on welfare over time. Fill in the following information, with units.

independent variable: ___________________ dependent variable: ______________

slope ______________ y-intercept _____________ linear function:

d) Use your function to estimate the number of people on welfare in the year 2003.