Scatter Plots and Regression Word Problems

Other things being equal, larger automobile engines are less fuel-efficient. You are planning

an experiment to study the effect of engine size (in liters) on the fuel efficiency (in miles per

gallon) of sport utility vehicles. In this study,

• gas mileage is a response variable, and you expect to find a negative association.

• gas mileage is a response variable, and you expect to find a positive association.

• gas mileage is an explanatory variable, and you expect to find a strong negative

  association.

• gas mileage is an explanatory variable, and you expect to find a strong positive

  association.

• gas mileage is an explanatory variable, and you expect to find very little association.

In a statistics course, a linear regression equation was computed to predict the final-exam score from the score on the first test. The equation was yˆ =10 + 0.9x where y is the final exam score and x is the score on the first test. Carla scored 95 on the first test. What is the

predicted value of her score on the final exam?

• 85.5

• 90

• 95

• 95.5

• None of these

In a statistics course, a linear regression equation was computed to predict the final-exam score from the score on the first test. The equation was yˆ =10 + 0.9x where y is the final exam score and x is the score on the first test. Carla scored 95 on the first test.

In the course ,Bill scored a 90 on the first test and a 93 on the final exam.

What is the value of his residual?

• -2

• 2

• 3

• 93

• None of these

The correlation between the heights of fathers and the heights of their (fully grown) sons is

r = 0.52. This value was based on both variables being measured in inches. If fathers'

heights were measured in feet (one foot equals 12 inches), and sons' heights were measured

in furlongs (one furlong equals 7920 inches), the correlation between heights of fathers and

heights of sons would be

• much smaller than 0.52

• slightly smaller than 0.52

• unchanged: equal to 0.52

• slightly larger than 0.52

• much larger than 0.52

All but one of the following statements contains an error. Which statement could be

correct?

• There is a correlation of 0.54 between the position a football player plays and his weight.

• We found a correlation of r = –0.63 between gender and political party preference.

• The correlation between the distance travelled by a hiker and the time spent hiking is

  r = 0.9 meters per second.

• We found a high correlation between the height and age of children: r = 1.12.

• The correlation between mid-August soil moisture and the per-acre yield of tomatoes is

  r = 0.53.

A set of data describes the relationship between the size of annual salary raises and the

performance ratings for employees of a certain company. The least squares regression

equation is

y-hat = 1400 + 2000x where y is the raise amount (in dollars) and x is the

performance rating. Which of the following statements must be true?

• For each one-point increase in performance rating, the raise will increase on average by

  $1400.

• The actual relationship between salary raises and performance rating is linear.

• The residuals for half the observations in the dataset will be positive.

• The correlation between salary raise and performance rating is negative.

• If the mean performance rating is 1.2, then the mean raise is $3800.

A least-squares regression line for predicting weights of basketball players on the basis of

their heights produced the residual plot below.

What does the residual plot tell you about the linear model?

• A residual plot is not an appropriate means for evaluating a linear model.

• The curved pattern in the residual plot suggests that there is no association between the

  weight and height of basketball players.

• The curved pattern in the residual plot suggests that the linear model is not appropriate.

• There are not enough data points to draw any conclusions from the residual plot.

• The linear model is appropriate, because there are approximately the same number of

  points above and below the horizontal line in the residual plot.

One concern about the depletion of the ozone layer is that the increase in ultraviolet (UV) light

will decrease crop yields. An experiment was conducted in a green house where soybean plants

were exposed to varying levels of UV, measured in Dobson units. At the end of the experiment

the yield (kg) was measured. A regression analysis was performed with the following results:

The least-squares regression line is the line that

• minimizes the sum of the distances between the actual UV values and the predicted UV

  values.

• minimizes the sum of the squared residuals between the actual yield and the predicted

  yield.

• minimizes the sum of the distances between the actual yield and the predicted UV.

• minimizes the sum of the squared residuals between the actual UV reading and the

  predicted UV values.

• minimizes the perpendicular distance between the regression line and each data point.

One concern about the depletion of the ozone layer is that the increase in ultraviolet (UV) light

will decrease crop yields. An experiment was conducted in a green house where soybean plants

were exposed to varying levels of UV, measured in Dobson units. At the end of the experiment

the yield (kg) was measured.

Which of the following is correct?

• If the UV value increases by 1 Dobson unit, the yield is expected to increase by 0.0463

  kg.

• If the yield increases by 1 kg, the UV value is expected to decrease by 0.0463 Dobson

  units.

• If the UV value increases by 1 Dobson unit, the yield is expected to decrease by 0.0463

  kg.

• The predicted yield is 4.3 kg when the UV value is 20 Dobson units.

• None of the above is correct.

Which statements below about least-squares regression are correct?

I. Switching the explanatory and response variables will not change the least-squares regression line.

II. The slope of the line is very sensitive to outliers in the x direction with large residuals.

III. A value of r2 close to 1 does not guarantee that the relationship between the variables is linear.

• Only I is correct.

• Only II is correct.

• Only III is correct.

• Both II and III are correct.

• All three statements—I, II, and III—are correct.