Appendix: Making and Using Graphs

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APPENDIX CHECKLIST After you have completed the appendix, you will have thoroughly reviewed the graphs used in your economics course.

Making and using graphs.

Graphs represent quantities as distances. The vertical axis is the y-axis and the horizontal axis is the xaxis. A scatter diagram plots a graph of one variable against the value of another variable. A timeseries graph measures time along the x-axis and the variable (or variables) of interest along the y-axis. A cross-section graph shows the values of an economic variable for different groups in the population at a point in time. Graphs can show the relationship between two variables in an economic model. Variables that move in the same direction have a positive, or direct, relationship. Variables that move in the opposite direction have a negative, or inverse, relationship. Some relationships have minimum or maximum points. The slope of a relationship is the change in the value of the variable measured on the y-axis divided by the change in the value of the variable measured on the x-axis. Using the symbol “Δ” to mean “change in”, the slope of a relationship equals Δy ÷ Δx. To graph a relationship among more than two variables, we use the ceteris paribus assumption and graph the relationship between two of the variables, holding the other variables constant.

CHECKPOINT 1  Making and using graphs. Additional Practice Problems 1. You have data on the average monthly rainfall and the monthly expenditure on umbrellas in Seattle, Washington. What sort of graph would be the best to reveal if any relationship exists between these variables? 2. In Figure A1.1, draw a straight line showing a positive relationship and another straight line showing a negative relationship.

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Part 1 . INTRODUCTION

Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Price (dollars per gallon) 1.12 1.22 1.56 1.53 1.44 1.64 1.92 2.34 2.64 2.85 3.32

3. The table has the average price of a gallon of gasoline, including taxes, for eleven years. In Figure A1.2, measuring years along the horizontal axis, label the axes and then plot these data. What type of graph are you creating? What is the general trend of gas prices during this decade?

4. Figure A1.3 shows the relationship between the price of a paperback book and the quantity of paperback books a publisher is willing to sell. What is the slope of the line in Figure A1.3?

Solutions to Additional Practice Problems 1 1. A scatter diagram would be the best graph to use. A scatter diagram would plot the monthly value of, say, rainfall along the vertical axis (the y-axis) and the monthly value of umbrella expenditure along the horizontal axis (the xaxis).

2. Figure A1.4 has two lines, one showing a positive relationship and the other showing a negative relationship. Your figure does not need to have identical lines. The key point your figure needs is that the line for the positive relationship slopes up as you move rightward along it and the line for the negative relationship slopes down as you move rightward along it.

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Appendix 1 . Making and Using Graphs

3. Figure A1.5 labels the axes and plots the data in the table. The graph is a time-series graph. The trend is positive because gas prices generally increased during these years.

4. The slope of a line is the change the variable measured on the y-axis divided by the change in the variable measured on the x-axis. To calculate the slope of the line in the figure, use points a and b in Figure A1.6. Between a and b, y rises by 2, from 4 to 6. And x increases by 400, from 200 to 600. The slope equals 2/400 = 0.005.

 Self Test 1 Fill in the blanks In a graph, the vertical line is called the ____ (xaxis; y-axis) and the horizontal line is called the ____ (x-axis; y-axis). A ____ (scatter diagram;

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time-series graph; cross-section graph) is a graph of the value of one variable against the value of another variable. A ____ (scatter diagram; time-series graph; cross-section graph) measures time along the x-axis and the variable along the y-axis. A ____ (scatter diagram; timeseries graph; cross-section graph) shows the values of an economic variable for different groups in the population at a point in time. If the graph of a relationship between two variables slopes up to the right, the two variables have a ____ (positive; negative) relationship. If the graph between two variables is a vertical line, the two variables ____ (are; are not) related. The slope of a relationship is the change in the value of the variable measured along the ____ (x-axis; y-axis) divided by the change in the value of the variable measured along the ____ (x-axis; y-axis). By using the ceteris paribus assumption, it ____ (is; is not) possible to graph a relationship that involves more than two variables. True or false 1. A point that is above and to the right of another point will have a larger value of the xaxis variable and a larger value of the y-axis variable. 2. A scatter diagram shows the values of an economic variable for different groups in a population at a point in time. 3. A time-series graph compares values of a variable for different groups at a single point in time. 4. A trend is a measure of the closeness of the points on a graph. 5. A positive relationship is always a linear relationship. 6. A relationship that starts out sloping upward and then slopes downward has a maximum. 7. A graph that shows a horizontal line indicates variables that are unrelated. 8. The slope at a point on a curve can be found by calculating the slope of the line that touches the point and no other point on the curve.

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Part 1 . INTRODUCTION

Multiple choice 1. Demonstrating how an economic variable changes from one year to the next is best illustrated by a a. scatter diagram. b. time-series graph. c. linear graph. d. cross-section graph. e. trend-line. 2. To show the values of an economic variable for different groups in a population at a point in time, it is best to use a a. scatter diagram. b. time-series graph. c. linear graph. d. cross-section graph. e. trend diagram. 3. If whenever one variable increases, another variable also increases, then these two variables are ____ related. a. positively b. negatively c. inversely d. cross-sectionally e. not

6. In figure A1.7, between points A and B, what is the slope of the line? a. 12 b. 3 c. 9 d. –9 e. 0

4. A graph of the relationship between two variables is a line that slopes down to the right. These two variables are ____ related. a. positively b. directly c. negatively d. not e. trend-line 5. Two variables are unrelated if their graph is i. a vertical line. ii. a 45 degree line. iii. a horizontal line. a. i only. b. ii only c. iii only d. i and iii. e. i, ii, and iii.

7. In Figure A1.8, an increase in z leads to a a. movement up along one of the lines showing the relationship between x and y. b. movement down along one of the lines showing the relationship between x and y. c. rightward shift of the line showing the relationship between x and y. d. leftward shift of the line showing the relationship between x and y. e. trend change in both x and y.

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Appendix 1 . Making and Using Graphs

8. In Figure A1.8, ceteris paribus, an increase in x is associated with a. an increase in y. b. a decrease in y. c. an increase in z. d. a random change in z. e. no change in either y or z. Complete the graph Year 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Workers (millions) 7.9 8.1 8.3 8.4 8.5 8.7 9.0 9.2 9.5 9.7

1. The table above gives the number of people working in restaurants and bars in the United States during 10 previous years.

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b. Using your figure, what was the trend in the number of people working in restaurants and bars during these years? Year 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Revenue (billions of dollars) 285 306 318 332 350 372 393 418 438 453

Workers (millions) 7.9 8.1 8.3 8.4 8.5 8.7 9.0 9.2 9.5 9.7

2. The table above gives the annual revenue for restaurants and bars and the number of people employed in restaurants and bars in the United States during 10 previous years. In Figure A1.10, measure the revenue along the horizontal axis and the number of workers along the vertical axis and plot the data. a. What type of graph are you creating? b. What relationship do you see in your figure between the revenue and the number of workers?

In Figure A1.9, measure time on the horizontal axis and the number of workers on the vertical axis, and then plot these data. a. What type of graph are you creating?

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Part 1 . INTRODUCTION

Price (dollars per sack of cat food) 1 2 3 4

Quantity (sacks of cat food per month) 10,000 8,000 7,000 4,000

5. In Figure A1.13, draw a line through point A with a slope of 2. Label the line “1.” Draw another line through point A with a slope of –2. Label this line “2.”

3. The number of sacks of premium cat food that cat lovers will buy depends on the price of a sack of cat food. The relationship is given in the table above. In Figure A1.11, plot this relationship, putting the price on the vertical axis and the quantity on the horizontal axis.

x 1 2 3 4

y 4 3 1 0

z 0 2 6 8

a. If the price of a sack of cat food is $2, how many sacks will be purchased? b. If the price of a sack of cat food is $3, how many sacks will be purchased? c. Is the relationship between the price and the quantity positive or negative? 4. In Figure A1.12, label the maximum and minimum points.

6. The table above contains data for three variables. a. In Figure A1.14, put y on the vertical axis and x on the horizontal axis. Show the relationship between x and y. Is this relationship positive or negative? b. What is the slope between x =2 and x = 3? c. In Figure A1.15 (on the next page), put z on the vertical axis and x on the horizontal axis. Show the relationship between x and z. Is this relationship positive or negative?

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Appendix 1 . Making and Using Graphs

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tween the number of DVDs purchased and the price when Bobby’s income is low. b. On the same figure, draw the relationship between the number of DVDs purchased and the price when his income is high. c. Does an increase in Bobby’s income shift the relationship between the price of a DVD and the number of DVDs purchased rightward or leftward?

Price (dollars per DVD) 11 12 13 14

Quantity of DVDs purchased, low income 4 3 1 0

Quantity of DVDs purchased, high income 5 4 3 2

7. Bobby says that he buys fewer DVDs when the price of a DVD is higher. Bobby also says that he will buy more DVDs after he graduates and his income is higher. The table above shows the number of DVDs Bobby buys in a month at different prices when his income is low and when his income is high.

Short answer and numeric questions 1. What are the three types of graphs? 2. If two variables are positively related, will the slope of a graph of the two variables be positive or negative? If two variables are negatively related, will the slope of a graph of the two variables be positive or negative? 3. If a line slopes upward to the right, is its slope positive or negative? If a line slopes downward to the right, is its slope positive or negative? 4. In Figure A1.17, what is the slope of the curved line at point A? At point B?

a. In Figure A1.16, put the price on the vertical axis and the quantity purchased on the horizontal axis. Show the relationship be-

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Part 1 . INTRODUCTION

SELF TEST ANSWERS  CHECKPOINT 1 Fill in the blanks In a graph, the vertical line is called the y-axis and the horizontal line is called the x-axis. A scatter diagram is a graph of the value of one variable against the value of another variable. A time-series graph measures time along the xaxis and the variable along the y-axis. A crosssection graph shows the values of an economic variable for different groups in the population at a point in time. If the graph of a relationship between two variables slopes up to the right, the two variables have a positive relationship. If the graph between two variables is a vertical line, the two variables are not related. The slope of a relationship is the change in the value of the variable measured along the y-axis divided by the change in the value of the variable measured along the x-axis. By using the ceteris paribus assumption, it is possible to graph a relationship that involves more than two variables.

Complete the graph

1. Figure A1.18 plots the data. a. This is a time-series graph; page 22. b. The trend is positive. During these 10 years there is an increase in the number of people working in restaurants and bars; page 22.

True or false 1. True; page 21 2. False; page 22 3. False; page 22 4. False; page 22 5. False; page 24 6. True; page 26 7. True; page 26 8. True; page 27 Multiple choice 1. b; page 22 2. d; page 22 3. a; page 24 4. c; page 25 5. d; page 26 6. d; page 27 7. d; page 28 8. b; page 28

2. Figure A1.19 plots the data. a. The figure is a scatter diagram; page 22. b. The relationship between the revenue and the number of workers is positive; page 24.

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Appendix 1 . Making and Using Graphs

3. Figure A1.20 plots the relationship. a. If the price is $2 per sack, 8,000 sacks are purchased; page 21. b. If the price is $3 per sack, 7,000 sacks are purchased; page 21. c. The relationship between the price and quantity of sacks is negative; page 25.

4. Figure A1.21 labels the two maximum points and one minimum point; page 26.

5. Figure A1.22 shows the two lines; page 27.

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6. a. Figure A1.23 plots the relationship. The relationship is negative; page 25. b. The slope equals (3 – 1) ÷ (2 – 3), which is −2; page 27.

c. Figure A1.24 plots the relationship. The relationship is positive; page 24.

7. a. Figure A1.25 plots the relationship; page 28.

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Part 1 . INTRODUCTION

b. Figure A1.25 plots the relationship; page 28. c. An increase in Bobby’s income shifts the relationship rightward; page 28. Short answer and numeric questions 1. The three types of graphs are scatter diagram, time-series graph, and cross-section graph; page 22. 2. If two variables are positively related, a graph of the relationship will have a positive slope. If two variables are negatively related,

a graph of the relationship will have a negative slope; pages 24, 25, 27. 3. If a line slopes upward to the right, its slope is positive. If a line slopes downward to the right, its slope is negative; page 27. 4. The slope of a curved line at a point equals the slope of a straight line that touches that point and no other point on the curve. The slope of the curved line at point A is –20 and the slope of the curved line at point B is 10; page 27.

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