1. A silly psychology student gathers data on the shoe size of 30 of his classmates and their GPA’s. The correlation coefficient between these two variables is most likely to be A) exactly –1.0 B) near -0.6 C) near 0 D) near +0.6 E) exactly +1.0
___ 2. A researcher studied the relationship between family income and amount of money spent on an automobile. She calculated that R2 = 45% . Which is the correct interpretaion? A) 45% of the price of the car can be predicted by using income.
B) The probability of predicting the correct price of a car is 45%.
C) 45% of the variability in car price can be explained by using income.
D) The car price fluctuates 45% more than income.
E) None of these
___ 3. If r = -0.4 for the relationship between the time of day and amount of coffee in an office worker’s mug, which are true?
I. r2 =-16%
II. There is a linear relationship between time and amount of coffee. III. 16% of the variability is correctly predicted by time of day.
A) I B) II C) III D) II and III only E) none of these
___ 4. The relationship between the longevity of an animal’s life and its gestation time is 0.70. If an animal is one standard deviation below average in life expectancy, the gestation time is predicted to be __?__ below average.
A) 1.4 SD B) 1 SD C) 0.7 SD D) 0.49 SD E) none of these
___ 5. We can use the length of a man’s hand span to predict his height, with a correlation coefficient of r = 0.60. If change our measurements from cm to m, the new correlation will be
A) 0.006 B) 0.06 C) 0.60 D) 6 E) none of these
___ 6. If a data set has a relationship that is best described by a linear model, than the residual plot will
A) also have a linear pattern with a similar correlation.
B) have no pattern with a correlation near 0.
C) have a curved pattern, like a parabola.
D) be an unknown shape.
E) none of these
___ 7. A regression model examining the amount of distance a long distance runner runs (in miles) to predict the amount of fluid the runner drinks (ounces) has a slope of 4.6. Which interpretation is appropriate?
A) The correlation is needed to interpret this value.
B) A runner drinks a minimum of 4.6 oz.
C) We predict 4.6 miles for every ounce that is drunk.
D) We predict for every mile run, the runner drinks 4.6 more ounces.
E) Each mile adds 4.6 more ounces.
___ 8. A regression equation is found that predicts the increased cost of a home owner’s electricity bill given the number of holiday lights they put on the outside of their house. The equation is dollars?= 2.5+0.02(light). If a house has 400 lights and a $15 increase in their electricity cost, find their residual.
A) -$15 B) -$5 C) $5 D) $15 E) $20
___ 9. Computer output in the scenario described in problem #8 reports that s = 2.3. Which is the correct interpretation of this value?
A) The slope of the regression line is $2.30 per light.
B) The slope of the regression line is 2.3 lights per dollar.
C) The correlation is 2.3.
D) The initial cost, even with no lights is $2.30
E) The average prediction error of the regression line is $2.30.
___ 10. Using the equation in number #8 again, if a homeowner doubles the number of lights he uses from 500 to 1000, how much do we predict he will increase his electric bill by?
A) $2 B) $10 C) $12.50 D) $22.50 E) $35
The Correct Answer and Explanation is :
Here are the correct answers for each question, along with detailed explanations:
1. The correlation coefficient between shoe size and GPA is most likely to be:
Answer: C) near 0
Explanation: In this case, the relationship between shoe size and GPA is likely to be weak or nonexistent. Shoe size and GPA are not typically correlated, meaning the correlation coefficient is near zero, indicating no relationship between the two variables.
2. If R² = 45% for the relationship between family income and money spent on an automobile, the correct interpretation is:
Answer: C) 45% of the variability in car price can be explained by using income.
Explanation: R² represents the proportion of variance in the dependent variable (car price) that can be explained by the independent variable (income). In this case, 45% of the variability in car price can be explained by family income.
3. If r = -0.4 for the relationship between time of day and coffee amount, which are true?
Answer: C) III. 16% of the variability is correctly predicted by time of day.
Explanation: r² = (-0.4)² = 0.16, meaning 16% of the variability in coffee consumption can be predicted by the time of day. The relationship is not linear (II is false) because of the negative correlation, but the percentage of variability is accurately explained by time of day.
4. If the correlation between longevity and gestation time is 0.70, and an animal is one standard deviation below average in life expectancy, the gestation time is predicted to be:
Answer: C) 0.7 SD
Explanation: The correlation coefficient (r) represents the proportion of how the two variables move together. If an animal is 1 SD below average in life expectancy, we predict the gestation time to be 0.7 SD below average, because r = 0.70.
5. If the correlation coefficient between hand span and height is r = 0.60, and measurements are converted from cm to meters, the new correlation will be:
Answer: C) 0.60
Explanation: Changing the unit of measurement (from cm to meters) does not affect the correlation coefficient, so it remains the same at 0.60.
6. If a relationship is best described by a linear model, the residual plot will:
Answer: B) have no pattern with a correlation near 0.
Explanation: A residual plot of a linear model should show no systematic pattern, meaning the residuals are randomly scattered around zero. If there’s a pattern, it suggests that a linear model is not appropriate.
7. If the slope of a regression model for predicting fluid intake based on running distance is 4.6, the appropriate interpretation is:
Answer: D) We predict for every mile run, the runner drinks 4.6 more ounces.
Explanation: The slope represents the change in the dependent variable (fluid intake) for each unit change in the independent variable (distance run). A slope of 4.6 means that for every additional mile run, the runner drinks 4.6 more ounces of fluid.
8. The regression equation predicting electricity costs is dollars = 2.5 + 0.02 × lights. If a house has 400 lights and a $15 increase in their electricity cost, the residual is:
Answer: C) $5
Explanation: Predicted increase in cost = 2.5 + 0.02 × 400 = $12.5. The residual is the actual value minus the predicted value, so the residual = $15 – $12.5 = $5.
9. The standard deviation of residuals (s = 2.3) in the regression model for predicting electricity costs means:
Answer: E) The average prediction error of the regression line is $2.30.
Explanation: The value of s (2.3) represents the standard deviation of the residuals, meaning the average amount by which the predictions of the regression model deviate from the actual values is $2.30.
10. Using the regression equation from question #8, if a homeowner doubles the number of lights from 500 to 1000, the predicted increase in the electric bill is:
Answer: D) $22.50
Explanation: Predicted increase in cost for 500 lights = 2.5 + 0.02 × 500 = $12.5. Predicted increase in cost for 1000 lights = 2.5 + 0.02 × 1000 = $22.5. The difference is $22.50 – $12.50 = $10.
These answers and explanations provide a deeper understanding of the interpretation and analysis of correlation and regression in statistics.