In the linear equation y = 4 + 3x, what is the slope?
3
✓
4
7
12
Correct Answer
3
In the form y = a + bx, the slope is b, the number multiplying x, so the slope is 3. The value 4 is the y-intercept a. The value 7 is the value of y when x = 1, not the slope. The value 12 multiplies the two constants, which has no meaning here.
Category: Linear Equations
Question 2
A tutoring company charges according to y = 25 + 40x, where x is the number of hours and y is the cost in dollars. What does the 25 represent?
The cost for each hour
A fee charged even for 0 hours
✓
The total cost of 40 hours
The hours bought for 40 dollars
Correct Answer
A fee charged even for 0 hours
The 25 is the y-intercept, the value of y when x = 0, so it is a fixed fee paid no matter how many hours are used. The cost for each hour is the slope, 40. The total cost of 40 hours would be 25 + 40(40) = 1,625 dollars. The hours bought for 40 dollars mixes up which number is a rate and which is a fee.
Category: Linear Equations
Question 3
A line passes through the points (1, 5) and (3, 11). What is its slope? Round to two decimal places if needed.
2
6
3
✓
0.33
Correct Answer
3
Slope is the change in y divided by the change in x: (11 - 5) / (3 - 1) = 6 / 2 = 3. The value 2 is only the change in x. The value 6 is only the change in y. The value 0.33 divides the change in x by the change in y, which is upside down.
Category: Linear Equations
Question 4
A scatter plot of weekly hours of exercise against resting heart rate shows points falling roughly along a straight line that slopes downward from left to right. What does this pattern show?
A positive association
No association
A curved association
A negative association
✓
Correct Answer
A negative association
When points trend downward from left to right, larger x values go with smaller y values, which is a negative association. A positive association would trend upward. No association would show no trend at all. A curved association would bend rather than follow a straight line.
Category: Scatter Plots
Question 5
Which of these is a possible value of the correlation coefficient r?
-0.85
✓
1.20
-1.5
2
Correct Answer
-0.85
The correlation coefficient always falls between -1 and 1, so -0.85 is possible and shows a fairly strong negative linear relationship. The value 1.20 is above the maximum of 1. The value -1.5 is below the minimum of -1. The value 2 is far above the maximum of 1.
Category: Correlation
Question 6
Which correlation coefficient shows the strongest linear relationship?
0.75
-0.91
✓
0.05
-0.40
Correct Answer
-0.91
Strength depends on how close |r| is to 1, and the sign shows only direction, so -0.91, with |r| = 0.91, is the strongest. The value 0.75 is strong but farther from 1. The value 0.05 shows almost no linear relationship. The value -0.40 is a weak to moderate negative relationship.
Category: Correlation
Question 7
A regression of final exam score on hours studied has r = 0.8. What percent of the variation in final exam scores is explained by the regression on hours studied?
80%
36%
64%
✓
8%
Correct Answer
64%
The coefficient of determination is r^2 = (0.8)^2 = 0.64, so 64% of the variation in scores is explained by the line. The value 80% uses r instead of r^2. The value 36% is the unexplained share, 1 - 0.64. The value 8% misplaces the decimal point.
Category: Correlation
Question 8
The regression line y-hat = 50 + 4x predicts exam score from x, the number of hours studied. What does the 4 tell you?
The predicted score for 0 hours of study
The correlation between hours and score
The share of score variation explained
The predicted gain in score per extra hour
✓
Correct Answer
The predicted gain in score per extra hour
The slope gives the predicted change in y for each one-unit increase in x, so each extra hour of study adds about 4 points to the predicted score. The predicted score for 0 hours of study is the intercept, 50. The correlation between hours and score is r, which is not shown in the equation. The share of score variation explained is r^2, not the slope.
Category: Least-Squares Line
Question 9
Using the regression line y-hat = 50 + 4x, where x is hours studied, what is the predicted exam score for a student who studies 6 hours?
74
✓
54
300
24
Correct Answer
74
Substitute x = 6: y-hat = 50 + 4(6) = 50 + 24 = 74. The value 54 adds the slope to the intercept without multiplying by 6. The value 300 multiplies the intercept by 6. The value 24 leaves out the intercept of 50.
Category: Prediction
Question 10
With the line y-hat = 50 + 4x, a student who studied 5 hours actually scored 64. What is this student's residual?
6
-6
✓
70
134
Correct Answer
-6
The predicted score is 50 + 4(5) = 70, and the residual is observed minus predicted, 64 - 70 = -6, so the point lies below the line. The value 6 subtracts in the wrong order, predicted minus observed. The value 70 is the predicted score, not the residual. The value 134 adds the observed and predicted scores.
Category: Least-Squares Line
Question 11
What quantity does the least-squares regression line make as small as possible?
The sum of the residuals
The number of points off the line
The sum of the squared residuals
✓
The largest single residual
Correct Answer
The sum of the squared residuals
The least-squares line is the line with the smallest possible sum of squared residuals, also called SSE. The sum of the residuals is always zero for this line, so it is not what defines it. The number of points off the line is not used, since most points fall off any line. The largest single residual is not the criterion either.
Category: Least-Squares Line
Question 12
For a data set, x-bar = 10, y-bar = 30 and the least-squares slope is b = 2. What is the y-intercept a?
30
50
-10
10
✓
Correct Answer
10
The least-squares line passes through (x-bar, y-bar), so a = y-bar - b(x-bar) = 30 - 2(10) = 10. The value 30 is just y-bar. The value 50 adds b(x-bar) instead of subtracting it. The value -10 subtracts in the wrong order, b(x-bar) - y-bar.
Category: Least-Squares Line
Question 13
A data set has correlation r = 0.5, standard deviation of x equal to 2 and standard deviation of y equal to 8. What is the slope of the least-squares line?
2
✓
0.125
4
8
Correct Answer
2
The slope is b = r(sy / sx) = 0.5(8 / 2) = 0.5(4) = 2. The value 0.125 flips the ratio to sx / sy. The value 4 is sy / sx without multiplying by r. The value 8 is just the standard deviation of y.
Category: Least-Squares Line
Question 14
A sample of 20 pairs is used to test whether the population correlation is zero. How many degrees of freedom does the t distribution for this test have?
19
18
✓
20
38
Correct Answer
18
The significance test for the correlation coefficient uses df = n - 2, so 20 pairs give 18 degrees of freedom. The value 19 uses n - 1, the rule for a one-sample t-test of a mean. The value 20 does not subtract anything. The value 38 counts all 40 individual numbers and subtracts 2.
Category: Significance Testing
Question 15
Which null hypothesis is used when testing whether a linear correlation exists in the population?
H0: r = 0
H0: rho = 1
H0: rho = 0
✓
H0: b = 1
Correct Answer
H0: rho = 0
Hypotheses are about population parameters, and the population correlation is rho, so H0: rho = 0 states that there is no linear relationship. H0: r = 0 uses the sample statistic r, which is already known. H0: rho = 1 claims a perfect relationship, which is not the claim being tested. H0: b = 1 concerns the sample slope and a value of 1 that has no special meaning here.
Category: Significance Testing
Question 16
A test of whether the population correlation is zero gives p = 0.01 with alpha = 0.05. Which conclusion fits?
Changes in x cause changes in y
The population correlation equals 0.01
No linear relationship has been shown
The data show a significant linear relationship
✓
Correct Answer
The data show a significant linear relationship
Since 0.01 is less than 0.05, H0 is rejected and there is sufficient evidence of a significant linear relationship between x and y. Changes in x cause changes in y goes too far, because correlation alone does not establish causation. The population correlation equals 0.01 confuses the p-value with rho. No linear relationship has been shown is the conclusion when p is larger than alpha.
Category: Significance Testing
Question 17
A regression of plant height on days of growth was built from data covering days 5 through 30. Using the line to predict height on day 90 is an example of what?
Extrapolation
✓
Interpolation
Residual analysis
Correlation
Correct Answer
Extrapolation
Extrapolation means predicting at an x value outside the range of the data, and day 90 is far beyond day 30, so the prediction may be unreliable. Interpolation is predicting inside the range of the data. Residual analysis examines the gaps between observed and predicted values. Correlation measures the strength of a linear relationship, not a prediction.
Category: Prediction
Question 18
The standard deviation of the residuals is s = 4. Using the rule that flags a point whose residual is at least 2s in absolute value, which residual marks a potential outlier?
-7
9
✓
6
-3
Correct Answer
9
Here 2s = 2(4) = 8, and only a residual of 9 has an absolute value of at least 8. The residual -7 has absolute value 7, which is under 8. The residual 6 is under 8. The residual -3 has absolute value 3, well under 8.
Category: Outliers
Question 19
A point lies far from the other points in the x direction, and removing it greatly changes the slope of the regression line. What is this point called?
A residual
A lurking variable
An influential point
✓
An intercept
Correct Answer
An influential point
An influential point is far from the rest of the data horizontally and can pull the regression line toward itself, so removing it noticeably changes the slope. A residual is a vertical distance from the line, not a kind of point. A lurking variable is an unmeasured third variable. An intercept is the value of y-hat at x = 0.
Category: Outliers
Question 20
A scatter plot shows a clear U-shaped pattern, and the correlation coefficient is close to 0. What is the best conclusion?
The variables are unrelated
The slope must be large
The data contain an error
The relationship is strong but not linear
✓
Correct Answer
The relationship is strong but not linear
The correlation coefficient measures only straight-line association, so a strong curved pattern can still give r near 0. The variables are unrelated is wrong because the U shape is a clear relationship. The slope must be large is wrong because b = r(sy/sx), so r near 0 makes the slope near 0. The data contain an error has no support, since curved patterns occur naturally.
Category: Correlation