Select an answer from the options below and click Submit.
Question 1
The data representing the cost of U.S. stamps appears to be linear. Perform a linear regression on the data. Which of the following would be the resulting residual plot?
Answer choice A is the linear regression residual plot. Even though the points are equally scattered above and below the horizontal axis, no pattern in the distribution of points is evident.
Answer choice B was an exponential regression residual plot. A definite pattern in the distribution of the points exists, and the points are unequally distributed above and below the x-axis.
Answer choice C was a quadratic regression residual plot. No definite pattern exists, and the distribution of points is not equal. Twice as many points are below the horizontal axis as above it.
Answer choice D was the original scatterplot.
Go to question 2.
Question 2
Shown below are the different residual plots that are formed from performing the various regressions on the United States stamp data.
| Linear Regression Residual Plot | Exponential Regression Residual Plot | Quadratic Regression Residual Plot |
|---|---|---|
 |
 |
 |
A linear regression is the best fit according to the residual plots. There is no pattern in the distribution of points, and the points are equally scattered above and below the horizontal axis.
Which correlation coefficient and coefficient of determination support this conclusion?
Answer choice A was the quadratic regression results.
Answer choice B was an exponential regression results.
Answer choice C is the linear regression results.
Answer choice D are the values from the quadratic regression but in reverse.
A quadratic regression produced the strongest correlation coefficient and coefficient of determination. Notice the residual plot is very similar to the linear residual plot. A person may incorrectly assume that a quadratic regression is a good fit. However, look closely at the quadratic regression’s residual plot. More of the points fall below the x-axis. In the linear regression, the points are equally distributed above and below the x-axis. This is why the linear fit is better than the quadratic fit. This example shows why residual plots are important. They are one more tool to use in regression analysis when determining curves (and lines) of best fit.
Go to question 3.
Question 3
The regression equation that best models the United States postage stamp data is a linear regression equation. If we assume that the cost of postage stamps will continue to rise at the same rate as the past forty years, we can use the regression equation, y = 0.913x − 1791.045, to determine the approximate year when a United States stamp will cost $1.00. In which year will this occur?
The data is in cents, so $1.00 would be 100 cents. The linear regression equation is y = 0.913x − 1791.045, where x is the year and y is the cost of the stamp in cents. Substitute 100 cents for y and solve for x. y = 0.913x − 1791.045
100 = 0.913x − 1791.045
1891.045 = 0.913x
2071.24 ≈ x y equals nine hundred thirteen thousandths x minus one thousand seven hundred ninety one and forty five thousandths. 100 equals nine hundred thirteen thousandths x minus one thousand seven hundred ninety one and forty five thousandths. One thousand eight hundred ninety one and forty five thousandths equals nine hundred thirteen thousandths x. Two thousand seventy one and twenty four hundredths is approximately the value of x. You also can enter the equation into your graphing calculator into Y1 and enter 100 into Y2, and find where the lines intersect.