Re-read IPS  Section 2.5 p. 181-188

Suppose a scatter plot of your data shows that the variables are correlated, but have a distinct curve. It is obviously NOT linear. Now what?

1. Data from the U.S. Office of Management and Budget show the United States national debt at the end of selected fiscal years. These data show that the national debt has been increasing dramatically since 1965. Can we predict the national debt for the year 2000?  National Debt Clock, updated daily.

United States National Debt

End of Fiscal Year             1965             1970             1975             1980             1985         1990         1995      2000
Amount (billions of $)          321                389              577               930              1946         3233         4974      5674

a. Sketch a scatter plot of the data.

b. Describe general shape of the graph.

c. Find the least squares regression line.

d. Graph the least squares line over the scatter plot.

e. Find the correlation.

f. Predict the national debt for the year 2000 based
on the least squares line.  Is it too high or too low?

g. Find the best-fit exponential regression equation. Graph.
STAT, CALC, ExpReg, enter the x-list, y-list, Y₂  (This pastes the regression equation into Y₂ so you can graph it.)

State the equation.

h. Find the correlation for the exponential equation.

i. Predict the national debt for the year 2000 based on the exponential equation.

j. Describe the difference between the predictions, the graphs, and the correlations.

2. A tour guide noticed that the larger the size of the group, the more time it took to assemble everyone for an event. The guide timed people and collected the following data.

Number of people                 2         3         4         5         6         7         8         9         10
Assembly time (in minutes)    2         2.6      3.4      4.4      5.7      7.4      9.7      12.5     16.3

a. Sketch a scatter plot of the data.

b. Find the linear regression model. Write the equation and graph it.

c. Find the exponential regression model. Write the equation and graph it.

d. Which of the two models, linear or exponential seems to fit the data better? Why?

3. A ball was thrown upward from an initial height of 15 m and the height of the ball was recorded every 0.5 seconds. The table shows the height h in meters t seconds after the ball was thrown (Func, Stat, & Trig, p. 119.)

time (seconds)   0.0         0.5         1.0         1.5         2.0         2.5         3.0         3.5         4.0         4.5
height (meters)   15           25.1      29.3       34.5       36.4       33.8       32.1        26         16.2       5.8

a. Sketch a scatterplot of the data.

b. Explain why neither a linear nor an exponential model fits these data.

c. What type of curve do the points seem to lie on?

d. Find the feature quadratic regression and use it to find an equation to model this data. State the equation.

e. What is the correlation coefficient for this model?

f. Plot the equation found in part (d) on the scatter plot.

g. The error is the difference between observed and predicted values.

Find the error in the quadratic regression model when t = 2.

h. Use <TRACE> to find the time(s) when the ball was 27 m off the ground.

Choosing a Good Model

4. The following are data on the amount of fertilizer applied to the soil, in pounds per square foot, and the yield of a certain food crop, in pounds per square yard (Freund & Simon, p. 520, #47).

fertilizer (lb/ft²)         .5         1.1         2.2         0.2         1.6         2.0
yield (lb/yd²)            32.0     34.3       15.7       20.8        33.5       21.5

a. Make a scatter plot of the data on your calculator.

b. Describe the distribution of the scatter plot.

c. Find the equation of best fit. Write it here.

d. Use the prediction equation to predict the yield when 1.5 pounds of the fertilizer are applied per square foot.

5. The following data pertain to the volume of a gas (in cubic inches) and its pressure (in pounds per square inch), when the gas is compressed at a constant temperature (Freund & Simon, p. 520 #46).

volume of gas (in³)     50         30         20         10         5
pressure (lb/in²)         16.0      40.1      78.0      190.5     532.2

a. Make a scatter plot of the data on your calculator.

b. Describe the distribution of the scatter plot.

c. Find the equation of best fit. Write it here.

d. Use the equation to estimate the pressure of this gas when it is compressed to a volume of 15 cubic inches.

e. Use the <TRACE> option to find the amount of fertilizer that provides the maximum yield.
 

6. Unemployment figures for the U.S. are given in the table below.

Year                                 1980         1985         1989         1990         1991         1992         1993         1994
Unemployed (thousands)   7637         8312         6528         6874         8426         9384         8734         7996

a. Make a scatter plot of the data on your calculator.

b. Describe the distribution of the scatter plot.

c. Which, if any, of a linear, exponential, or quadratic function seems to model these data? Justify your answer.