Scatterplots-Chap 3&12-MC

As shown above, a least-squares regression line has been fitted to the winning percentages for a local sports team in each of the years 1983 through 1995. The percentage for the 1996 season was then plotted (as circled above). Which of the following statements correctly describes how the value for the 1996 season will change the appearance of the least-squares regression line and the correlation coefficient if a new least-squares regression line is fitted to the 1983 through 1996 data?

• The 1996 point will make the least-squares regression line steeper and the correlation coefficient stronger.

• The 1996 point will make the least-squares regression line steeper and the correlation coefficient weaker.

• The 1996 point will make the least-squares regression line closer to horizontal and the correlation coefficient stronger.

• The 1996 point will make the least-squares regression line closer to horizontal and the correlation coefficient weaker.

• The 1996 point will not have any effect on the least-squares regression line since it follows the same downward trend.

Consider n pairs of numbers (x₁, y₁), (x₂, y₂), ..., and (xₙ, yₙ). The mean and standard deviation of the x-values are x̄ = 5 and sₓ = 4, respectively. The mean and standard deviation of the y-values are ȳ = 10 and sᵧ = 10, respectively. Of the following, which could be the least squares regression line?

• ŷ = -5.0 + 3.0x

• ŷ = 3.0x

• ŷ = 5.0 + 2.5x

• ŷ = 8.5 + 0.3x

• ŷ = 10.0 + 0.4x

A high school physics teacher was conducting an experiment with his class on the length of time it will take a marble to roll down a sloped chute. The class ran repeated trials in order to determine the relationship between the length, in centimeters, of the sloped chute and the time, in seconds, for the marble to roll down the chute. A linear relationship was observed and the correlation coefficient was 0.964. After discussing their results, the teacher instructed the students to convert all of the length measurements to meters but leave the time in seconds. What effect will this have on the correlation of the two variables?

• Because the standard deviation of the lengths in meters will be one hundredth of the standard deviation of the lengths in centimeters, the correlation will decrease by one hundredth to 0.954.

• Because the standard deviation of the lengths in meters will be one hundredth of the standard deviation of the lengths in centimeters, the correlation will decrease proportionally to 0.00964.

• Because changing from centimeters to meters does not affect the value of the correlation, the correlation will remain 0.964.

• Because only the length measurements have been changed, the correlation will decrease substantially.

• Because meters are a much more common measurement for length in determining speed, the linear relationship of the data will be stronger and thus the correlation will increase substantially.

The equation of the least squares regression line for the points on the scatterplot is y = 1.3 + 0.73x. What is the residual for the point (4, 7)?

• 2.78

• 3.00

• 4.00

• 4.22

• 7.00

A delivery service places packages into large containers before flying them across the country. These filled containers vary greatly in their weight. Suppose the delivery service’s airplanes always transport two such containers on each flight. The two containers are chosen so their combined weight is close to, but does not exceed, a specified weight limit. A random sample of flights with these containers is taken, and the weight of each of the two containers on each selected flight is recorded. The weights of the two containers on the same flight:

• will have a correlation of 0

• will have a negative correlation

• will have a positive correlation that is less than 1.

• will have a correlation of 1

• cannot be determined from the information given

In the scatterplot of y versus x shown above, the least squares regression line is superimposed on the plot. Which of the following points has the largest residual?

• A

• B

• C

• D

• E

Based on the computer output above, which of the following is the best interpretation of the value of the slope of the regression line?

• For each additional kilogram of lean body mass, the resting metabolic rate increases on average by 22.563 calories per day.

• For each additional kilogram of lean body mass, the resting metabolic rate increases on average by 264.0 calories per day.

• For each additional kilogram of lean body mass, the resting metabolic rate increases on average by 144.9 calories per day.

• For each additional calorie per day for the resting metabolic rate, the lean body mass increases on average by 22.563 kilograms.

• For each additional calorie per day for the resting metabolic rate, the lean body mass increases on average by 264.0 kilograms.

There is a linear relationship between the number of chirps made by the striped ground cricket and the air temperature. A least squares fit of some data collected by a biologist gives the model: y = 25.2 + 3.3x, 9 < x < 25, where x is the number of chirps per minute and y is the estimated temperature in degrees Fahrenheit. What is the estimated increase in temperature that corresponds to an increase of 5 chirps per minute?

• 3.3° F

• 16.5° F

• 25.2° F

• 28.5° F

• 41.7° F

Mr. Nerdly asked the students in his AP Statistics class to report their overall grade point averages and their SAT Math scores. The scatterplot below provides information about his students' data. The dark line is the least-squares regression line for the data, and its equation is ŷ = 410.54 + 67.3x. Which of the following statements about the circled point is false?

• This student has a grade point average of about 2.9 and an SAT Math score of about 690.

• If we used the least-squares line to predict this student’s SAT Math score, we would make a prediction that is too low.

• This student’s residual is negative.

• Removing this data point would cause the correlation coefficient to increase.

• Removing this student’s data point would increase the slope of the least-squares line.

Suppose we fit a least-squares regression line to a set of data. What is true if a plot of the residuals shows a curved pattern?

• A straight line is not a good model for the data.

• The correlation must be 0.

• The correlation must be positive.

• Outliers must be present.

• The regression line might or might not be a good model for the data, depending on the extent of the curve.