Title: Examining Relationships: Quantitative Data
Course: Question Banks

Q: The following research question describes a relationship between two quantitative variables. Which variable should be plotted on the horizontal X-axis?

How is the age of a teenager related to the average number of text messages they send and receive each day?
- average number of text messages (Average number of text messages is the response variable plotted on the Y-axis.)
- either (The research question implies that age might explain the quantity of text messaging. So there is a logical explanatory-response relationship between these variables.)
- age (Age is the explanatory variable plotted on the X-axis.)
A: age

Q: The following research question describes a relationship between two quantitative variables. Which variable should be plotted on the horizontal X-axis?

Is the sales price of a townhouse in San Francisco related to the number of square feet in the townhouse?
- sales price (The number of square feet is a measurement of the size of the townhouse. So the number of square feet might influence the sales price. Therefore, sales price is the response variable and plotted on the Y-axis.)
- either (There is a logical explanatory-response relationship between these variables. The number of square feet is a measurement of the size of the townhouse. So the number of square feet could partially explain the sales price.)
- square footage (Number of square feet is the explanatory variable plotted on the X-axis. Square footage could be used to predict sales price.)
A: square footage

Q: The following research question describes a relationship between two quantitative variables. Which variable should be plotted on the horizontal X-axis?

For women aged 25-35 what is the relationship between their annual salary and the number of years of education?
- salary (The salary is probably affected by the amount of education. So salary is the response variable and plotted on the Y-axis.)
- either (There is a logical explanatory-response relationship between these variables. The amount of education could partially explain a woman’s salary.)
- years of education (Years of education is the explanatory variable plotted on the X-axis. Amount of education could influence salary.)
A: years of education

Q: The following research question describes a relationship between two quantitative variables. Which variable should be plotted on the horizontal X-axis?

Is the amount of fat in a serving of cereal related to the amount of sugar in a serving?
- fat in a serving (There is no logical explanatory-response relationship between these variables. We might expect fat and sugar to be associated, but either one could be used to predict the other. So either variable could be plotted on the X-axis.)
- sugar in a serving (There is no logical explanatory-response relationship between these variables. We might expect fat and sugar to be associated, but either one could be used to predict the other. So either variable could be plotted on the X-axis.)
- either (There is no logical explanatory-response relationship between these variables. We might expect fat and sugar to be associated, but either one could be used to predict the other. So either variable could be plotted on the X-axis.)
A: either

Q: Choose the most likely correlation value for this scatterplot:
- r = 0.436 (You correctly identified that the direction of the relationship is positive. However, the linear relationship is not this strong. Please try again.)
- r = -0.897 (The direction of the relationship is positive. Please try again.)
- r = 0.995 (You correctly identified that the direction of the relationship is positive. However, the linear relationship is not this strong. Please try again.)
- r = -0.575 (The direction of the relationship is positive. Please try again.)
- r = 0.100 (The direction of the relationship is positive and the linear relationship is weak (close to zero).)
A: r = 0.100

Q: Choose the most likely correlation value for this scatterplot:
- r = 0.436 (The direction of the relationship is negative. Please try again.)
- r = 0.100 (The direction of the relationship is negative. Please try again.)
- r = 0.995 (The direction of the relationship is negative. Please try again.)
- r = -0.575 (Although the direction of the relationship is negative, the linear relationship is stronger. Please try again.)
- r = -0.897 (The direction of the relationship is negative and the linear relationship is quite strong.)
A: r = -0.897

Q: Choose the most likely correlation value for this scatterplot:
- r = 0.436 (Look again at the strength of the linear relationship. Please try again.)
- r = 0.100 (Look again at the strength of the linear relationship. Please try again.)
- r = -0.897 (Look again at the direction of the relationship. Please try again.)
- r = -0.575 (Look again at the direction and the strength of the relationship. Please try again.)
- r = 0.995 (The direction of the relationship is positive and the linear relationship is quite strong.)
A: r = 0.995

Q: Choose the most likely correlation value for this scatterplot:
- r = 0.436 (Look again at the direction of the linear relationship. Please try again.)
- r = 0.100 (Look again at the direction and the strength of the linear relationship. Please try again.)
- r = -0.897 (Look again at the strength of the linear relationship. Please try again.)
- r = 0.995 (Look again at the direction and the strength of the linear relationship. Please try again.)
- r = -0.575 (The direction of the relationship is negative, and the linear relationship is only moderately strong.)
A: r = -0.575

Q: Choose the most likely correlation value for this scatterplot:
- r = 0.100 (Look again at the strength of the linear relationship. Please try again.)
- r = -0.897 (Look again at the direction and the strength of the linear relationship. Please try again.)
- r = 0.995 (Look again at the strength of the linear relationship. Please try again.)
- r = -0.575 (Look again at the direction of the linear relationship. Please try again.)
- r = 0.436 (The direction of the relationship is positive and the linear relationship is only moderately strong.)
A: r = 0.436

Q: Two students conduct a study to investigate the relationship between forearm length and height.

Maria measures the subjects in centimeters. In a scatterplot of the data she sees a linear relationship between the variables, so she calculates the correlation coefficient. She determines that r = 0.86.

John measures the same subjects in inches. He also calculates the correlation coefficient. What do you expect the correlation will be for John’s measurements?
- John’s correlation will be approximately 0.86(0.61) = 0.52 because one centimeter = 0.61 inches. (A change in units will change the individual measurements but not the correlation. For example, we can convert a person’s height from cm to inches: a person who is 100 cm tall is 100(0.61) = 61 inches tall. But the pattern we see in the measurements will not change. Correlation measures how closely the data fits a linear pattern.)
- John’s correlation will be approximately r = 1 because there is a very strong relationship between John’s measurements and Maria’s measurements. (Correlation is a measure of the strength and direction of the linear relationship between the two variables: forearm length and height. We are not looking at the relationship between Maria’s measurements and John’s measurements.)
- John’s correlation will be approximately 0.86 because the pattern in the data will be the same. (A change in units does not change the correlation.)
A: John’s correlation will be approximately 0.86 because the pattern in the data will be the same.

Q: Which scatterplot has a correlation coefficient close to 0 but also has a strong relationship between the variables?
- Scatterplot 2 (You are correct that Scatterplot 2 has a correlation coefficient close to 0, but there is no pattern to suggest a strong relationship between the variables in Scatterplot 2. Try again.)
- Scatterplot 3 (You are correct that Scatterplot 3 has a fairly strong relationship. Since the relationship is linear with a positive association, the correlation coefficient is close to 1, not 0. Try again.)
- Scatterplot 1 (Scatterplot 1 has a strong curvilinear form with r close to 0. This is a good example of how the correlation coefficient does not give us information about the strength of relationships that are not lines.)
A: Scatterplot 1

Q: For a sample of cities in the U.S. the correlation between the murder rate and number of police officers employed by the city is r = 0.72. What can we conclude?
- There is a strong positive linear relationship between murder rate and size of the police force. (Recall that r does not give us reliable information about the form of the relationship. We cannot conclude that the relationship is linear.)
- Cities are responding to an increase in the murder rate by hiring more police officers. (This statement implies that the increase in the murder rate is causing the increase in size of the police force. A positive correlation between murder rate and number of police officers could be explained by other lurking variables, such as the size of the city.)
- The size of the city is a possible lurking variable explaining the association between murder rate and size of the police force. (Larger cities will tend to have higher murder rates and more police officers.)
A: The size of the city is a possible lurking variable explaining the association between murder rate and size of the police force.

Q: The scatterplot above plots number of unemployed Americans (in thousands) against gross national product (in billions of U.S. dollars). The data were gathered in an economics study from 1947 to 1962 by J.W. Longley.

(Reference: J. W. Longley (1967) An appraisal of least-squares programs from the point of view of the user. Journal of the American Statistical Association, 62, 819-841.)

The correlation is 0.60

What might we conclude from this data?
- The weak positive correlation shows that there is no relationship between unemployment numbers and gross national product. (A weak correlation between two variables might occur if there is a lurking variable.)
- The weak positive correlation shows that higher unemployment numbers cause lower gross national product. (A correlation between the two variables doesn’t mean that one causes the other.)
- Common sense suggests that unemployment numbers and gross national product are negatively correlated. There is likely a lurking variable . (A lurking variable may hide the true relationship between two variables.)
A: Common sense suggests that unemployment numbers and gross national product are negatively correlated. There is likely a lurking variable .

Q: The scatterplot above plots number of unemployed Americans (in thousands) against gross national product (in billions of U.S. dollars). The data were gathered in an economics study from 1947 to 1962 by J.W. Longley.

(Reference: J. W. Longley (1967) An appraisal of least-squares programs from the point of view of the user. Journal of the American Statistical Association, 62, 819-841.)

The correlation is 0.60

Which of the following is the most appropriate conclusion given this finding?
- Gross domestic product is a lurking variable. (GDP and GNP are similar metrics to estimate market value of goods.)
- A higher GNP causes higher unemployment numbers. (An association does not imply causation. We cannot conclude that either variable causes an increase in the other.)
- U.S. Population is a lurking variable. (Since the number of unemployed Americans is closely related to the total population (the larger a population often means a larger number of unemployed).)
A: U.S. Population is a lurking variable.

Q: For a statistics project a student gathered data on monthly car insurance premiums paid by students and faculty at a California community college in 2008. She found a fairly strong linear relationship between driving experience and monthly car insurance premiums.

She used the line shown in the scatterplot to summarize the pattern in the data.

Here is the equation of the line:

Predicted monthly car insurance premium = 97 – 1.45 * years of driving experience.

The predicted monthly car insurance premium paid by someone who has been driving 12 years is ________ dollars a month.
- 84.3 (Try plugging in 12 into the equation for years, multiply by -1.45 and add 97.)
- 89.5 (Try plugging in 12 into the equation for years, multiply by -1.45 and add 97.)
- 79.6 (Plug 12 into the equation. This gives you 97 - 1.45 * 12 = 79.6.)
A: 79.6

Q: A study was conducted to explore whether there is a relationship between the amount of protein and the amount of fat that people consume. In the study, volunteers kept a record of all the food they ate. The amount of protein (in grams per week) and the amount of fat (in grams per week) was then calculated for each person.

Here is a summary of the entire data set:

Furthermore, the correlation between fat and protein is r = 0.9224.

Here is a scatterplot of the data (let X = fat and Y = protein):

Would a linear regression give a reasonable prediction for the amount of protein a person eats?
- No, because there is too much variation in the data. (Although there is variation in the data, it is more important to look at the overall shape of the scatterplot.)
- No, because the scatterplot shows a curve, not a straight line. (The form of the relationship is linear.)
- Yes, because there is a positive association between protein and fat. (Although this conclusion is correct, it is more important to look at the shape of the scatterplot.)
- Yes, because the scatterplot shows a linear relationship. (The form of the relationship is linear.)
A: Yes, because the scatterplot shows a linear relationship.

Q: A study was conducted to explore whether there is a relationship between the amount of protein and the amount of fat that people consume. In the study, volunteers kept a record of all the food they ate. The amount of protein (in grams per week) and the amount of fat (in grams per week) was then calculated for each person.

Here is a summary of the entire data set:

Furthermore, the correlation between fat and protein is r = 0.9224.

Here is a scatterplot of the data (let X = fat and Y = protein):

What is the slope of the least-squares regression line?
- 0.9224 (0.9224 is the correlation. The correlation is related to the slope by the formula: slope = (r * s_y)/s_x.)
- 0.9039 (You may have interchanged X and Y. X is fat and Y is protein. The slope is (r * s_y) / s_x. Divide the standard deviation of protein by the standard deviation of fat.)
- 0.9413 (The slope is (r * s_y) / s_x = 0.9224 (69.8 / 68.4) = 0.9413.)
A: 0.9413

Q: A study was conducted to explore whether there is a relationship between the amount of protein and the amount of fat that people consume. In the study, volunteers kept a record of all the food they ate. The amount of protein (in grams per week) and the amount of fat (in grams per week) was then calculated for each person.

Here is a summary of the entire data set:

Furthermore, the correlation between fat and protein is r = 0.9224.

Here is a scatterplot of the data (let X = fat and Y = protein):

What is the Y-intercept of the least-squares regression line?
- 44.69 (You may have interchanged X and Y. X is fat and Y is protein. The Y-intercept is (mean for Y) - (slope)*(mean for X).)
- 46.06 (You may have used the correlation instead of the slope to calculate the Y-intercept. The Y-intercept is (mean for Y) - (slope)*(mean for X). The slope is 0.9413.)
- 33.44 (The Y-intercept is (mean for Y) - (slope)*(mean for X) = 662.5 - 0.9413(668.3).)
A: 33.44

Q: A study was conducted to explore whether there is a relationship between the amount of protein and the amount of fat that people consume. In the study, volunteers kept a record of all the food they ate. The amount of protein (in grams per week) and the amount of fat (in grams per week) was then calculated for each person.

Here is a summary of the entire data set:

Furthermore, the correlation between fat and protein is r = 0.9224.

Here is a scatterplot of the data (let X = fat and Y = protein):

What is the equation of the least-squares regression line?
- predicted fat = 33.43 + 0.9413 * protein (You have interchanged X and Y. X is fat and Y is protein. We are using fat to predict protein. You correctly located the slope (b) and the y-intercept (a) in the equation.)
- predicted fat = 0.9413 + 33.43 * protein (There are two mistakes here: (1) X and Y are interchanged. X is fat and Y is protein. We are using fat to predict protein. (2) The slope and the Y-intercept are interchanged. The equation is: predicted Y = Y-intercept + (slope)*X.)
- predicted protein = 0.9413 + 33.43 * fat (The slope and Y-intercept are interchanged in the equation. The equation is: predicted Y = Y-intercept + (slope)*X.)
- predicted protein = 33.43 + 0.9413 * fat (predicted Y = Y-intercept + (slope)*X in this context is predicted protein = 33.43 + (0.9413) * fat.)
A: predicted protein = 33.43 + 0.9413 * fat

Q: This scatterplot is based on data points that have a correlation of r =.91.

Which assessment of the fit of the linear model is the most appropriate?
- The pattern in the residual plot suggests that predictions based on the linear regression line will result in greater error as we move from left to right through the range of the explanatory variable. (The points on the residual plot stay about the same distance away from the horizontal axis as we move left-to-right.)
- The linear model is not appropriate because there is a distinctive curvilinear pattern in the residual plot. (The points in the residual plot of a linear model don’t seem to fall close to any smooth curve.)
- The linear model is appropriate because there is no pattern in the residual plot. (The residual plot shows no clear pattern, so the linear model is appropriate.)
A: The linear model is appropriate because there is no pattern in the residual plot.