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I was on the second day of a road trip when I decided to keep a record of how far I had traveled from home. The table below shows how many hours I drove that day and how far I was away from home. For example, after I had traveled a total of 3½ hours, I was 366 miles away from home. - PowerPoint PPT Presentation
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I was on the second day of a road trip when I decided to keep a record of how far I had traveled from home. The table below shows how many hours I drove that day and how far I was away from home. For example, after I had traveled a total of 3½ hours, I was 366 miles away from home.
Number of hours traveled
on day 2
Distance from home (in miles)
2 282
3½ 366
6 506
8 618
8½ 646
Without graphing, determine if the data in the table below are linear.
84
140
112
28
1.5
2.5
2
.5Conclusion: Since the slopes
between consecutive pairs of points are the same, the data are linear.
I was on the second day of a road trip when I decided to keep a record of how far I had traveled from home. The table below shows how many hours I drove that day and how far I was away from home. For example, after I had traveled a total of 3½ hours, I was 366 miles away from home.
Number of hours traveled
on day 2
Distance from home (in miles)
2 282
3½ 366
6 506
8 618
8½ 646
Without graphing, determine if the data in the table below are linear.
Exit Ticket
a.Find the linear model (equation) for the data.
b.Graph the data points and the model on your calculator. Call your teacher over to see your calculator screen.
c.How far was I from home at the start of the day?
y = 56x + 170
x = number of hoursy = distance in miles
170 miles
Answers to even-numbered HW problemsSection 3.3
S-2 Since x-values are evenly spaced and y-values are not, the data are not linear.
S-18 Calculator Graph
Ex 2a) y = 1445x – 2,870,269
b) Graph
c) The formula predicts a tuition of $34,181 for 2010.
Ex 12b) The slope is -.09. It means number of graduates
c) G = -.09y +181.48 where y = years and G = number of graduates in millions.
d) G(1994) = 2.02 million
x 2 4 6 8
y 12 17 21 25
5 4 4 Date Average Tuition
2001 $21,176
2002 $22,621
2003 $24,066
2004 $25,511
2005 $26,956
1
1
1
1
1445
1445
1445
1445
where x = years and y = tuition
The data are linear because all the dates increase by the same amount and all the tuitions increase by the same amount
Year # graduating (n millions)
1985 2.83
1987 2.65
1989 2.47
1991 2.29
decreased, on average, by .09 million (90,000) each year from 1985 to 1991.
2
2
2
.18
.18
.18
The data are linear because all the years increase by the same amount and the number of graduates decreases by the same amount
The data are linear because the slope between each pair of points is the same (-.09)
The chart below represents the number of automobiles (in millions) produced worldwide in each of the years listed.
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
1970 1975 1980 1985 1990
Use the graphing calculator to make a scatter plot of the data using years since 1970.
The chart below represents the number of automobiles (in millions) produced worldwide in each of the years listed.
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
1970 1975 1980 1985 1990
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
A = .65t + 23.3
A = # of automobiles produced (in millions)
t = number of years since 1970
Are the data approximately linear? Are the data linear?
What linear function would be a good model for this data?
y = .65x + 23.3
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
The Regression Line (also called the least-squares fit line or the best fit line) for a set of data is a line that is, on average, closest to each data point.
0 5 10 15 20Years since 1970
A = .65t + 23.3
A = # of automobiles produced (in millions)
t = number of years since 1970
y = .65x + 23.3
Year 0 5 10 15 20Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
Year 0 5 10 15 20Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
Year 0 5 10 15 20Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
Regression equation: y = .648x + 22.78
A = # of automobiles produced (in millions)
t = number of years since 1970
A = .648t + 22.78
A = .65t + 23.3
1970 1975 1980 1985 1990
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
Regression equation: y = .648x + 22.78
A = # of automobiles produced (in millions)
t = number of years since 1970
A = .648t + 22.78
Question: Explain in practical terms the meaning of the slope of the regression line model.
1970 1975 1980 1985 1990
The slope (.648) means that automobile production increased, on average, by .648 million each year from 1970 to 1990.
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
Regression equation: y = .648x + 22.78
A = # of automobiles produced (in millions)
t = number of years since 1970
A = .648t + 22.78
Question: What is the model’s estimate of automobile production in 1970? How far from the actual 1970 automobile production is the estimate?
1970 1975 1980 1985 1990
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
Regression equation: y = .648x + 22.78
A = # of automobiles produced (in millions)
t = number of years since 1970
A = .648t + 22.78
The model’s estimate for 1970 is 22.78 million cars produced. This differs from the actual figure by .52 million cars.
1970 1975 1980 1985 1990
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
Regression equation: y = .648x + 22.78
A = # of automobiles produced (in millions)
t = number of years since 1970
A = .648t + 22.78
Question: Use the regression equation to estimate automobile production in 2005.
1970 1975 1980 1985 1990
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
0 5 10 15 20Years since 1970
Regression equation: y = .648x + 22.78
A = # of automobiles produced (in millions)
t = number of years since 1970
A = .648t + 22.78
The model’s estimate for number of automobiles produced in 2005 is 45.46 million.
1970 1975 1980 1985 1990
Source: Worldwatch Institute http://www.worldwatch.org/node/4288
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
The model’s estimate for number of automobiles produced in 2005 is 45.46 million.
Source: Worldwatch Institute http://www.worldwatch.org/node/4288
“Global passenger car production grew in 2005 to 45.6 million units.”
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
Year 1970 1975 1980 1985 1990Automobile Production
(in millions)23.3 25.6 29.1 32.0 36.3
The model’s estimate for number of automobiles produced in 2005 is 45.46 million.
Homework:
Read Section 3.4 (through bottom of page 237)
Page 278 # S-1, S-3, S-5
Pages 280 – 282 # 1, 2, 5, 9
* All graphs to be done on the calculator
0
10
20
30
Year U.S. Population (in millions)
1960 180.7 1970 205.1 1980 227.7 1990 249.9
The table below shows the population of the United States in millions.
Graph the data on a graphing calculator using years since 1960.
0
10
20
30
Year U.S. Population (in millions)
1960 180.7 1970 205.1 1980 227.7 1990 249.9
The table below shows the population of the United States in millions.
1. Find the equation of the regression line using years since 1960.
2. Identify the slope of the regression line and explain what it means in this setting.
3. Use the regression equation to estimate the U.S. population in the year 2000.
0
10
20
30
Year U.S. Population (in millions)
1960 180.7 1970 205.1 1980 227.7 1990 249.9
1. Find the equation of the regression line using years since 1960.
2. Identify the slope of the regression line and explain what it means in this setting.
3. Use the regression equation to estimate the U.S. population in the year 2000.
y = 2.302x + 181.32
The slope is 2.302. It means that the U.S. population increased, on average, by 2.302 million people per year from 1960 to 1990.
Based on the model, the population of the U.S. in 2000 was 273.4 million.
where x = years since 1960 and y = population in millions.
The table below shows the population of the United States in millions.
0
10
20
30
Based on actual census figures, the actual U.S. population in 2000 was 281,421,906.
Year U.S. Population (in millions)
1960 180.7 1970 205.1 1980 227.7 1990 249.9
The table below shows the population of the United States in millions.
Based on the model, the population of the U.S. in 2000 was 273.4 million.
