AP Statistics Quiz: Residuals

What this quiz covers

This quiz focuses on Residuals, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Model B provides a better fit to the data than Model A because its sum of squared residuals is smaller. (correct answer)
2. Model A provides a better fit to the data than Model B because its sum of squared residuals is larger.
3. Neither model is a good fit, as the sum of squared residuals should be close to zero for any good fit.
4. Both models fit the data equally well; the choice depends on which model is simpler to interpret.

Explanation: The sum of squared residuals measures the total prediction error of a model. A model with a smaller sum of squared residuals provides a better fit to the data because the data points are, on average, closer to the model's predictions.

Question 2

1. The linear model is an appropriate way to describe the relationship between hours studied and test scores. (correct answer)
2. There is no statistical association between hours studied and test scores for these students.
3. The relationship between hours studied and test scores is a strong positive, but nonlinear, association.
4. The variance of the test scores is the same for all students, regardless of the hours they studied.

Explanation: A residual plot that shows a random scatter of points around the zero line with no obvious pattern indicates that a linear model is appropriate for the data.

Question 3

1. It minimizes the sum of the squared residuals. (correct answer)
2. It minimizes the number of points that are outliers.
3. It passes through the maximum number of data points.
4. It minimizes the sum of the residuals, which must be a non-negative value.

Explanation: The least-squares regression line is defined as the line that minimizes the sum of the squared vertical distances (residuals) from the data points to the line.

Question 4

1. The variability of the prediction errors is not constant across all levels of advertising spending. (correct answer)
2. The relationship between profits and advertising spending is curved rather than linear.
3. The linear model is generally appropriate because the residuals appear to be centered around zero.
4. The data contains an influential point at a high level of spending that increases the spread.

Explanation: A fan-shaped or cone-shaped pattern in a residual plot indicates that the variance of the residuals is not constant (a condition known as heteroscedasticity). This violates a key assumption for regression inference.

Question 5

1. The residual is calculated as $16,200 - 15,000$. (correct answer)
2. The residual is calculated as $15,000 - 16,200$.
3. The residual is calculated as $50,000 - 15,000$.
4. The residual is calculated as $16,200 - 50,000$.

Explanation: The residual is defined as the difference between the actual observed value (y) and the value predicted by the regression model ($\hat{y}$). In this case, the calculation is $y - \hat{y} = 16,200 - 15,000$.

Question 6

1. 0.8 cm (correct answer)
2. -0.8 cm
3. 6.2 cm
4. 5.8 cm

Explanation: First, calculate the predicted height using the regression equation: $\hat{y} = 1.2 + 0.5(10) = 1.2 + 5 = 6.2$ cm. The residual is the actual height minus the predicted height: $residual = y - \hat{y} = 7.0 - 6.2 = 0.8$ cm.

Question 7

1. The student's actual final exam score was 5 points higher than the score predicted by the model. (correct answer)
2. The student's actual final exam score was 5 points lower than the score predicted by the model.
3. The student's midterm exam score was 5 points higher than their predicted final exam score.
4. The model's prediction was off by an average of 5 points for the students in the sample.

Explanation: A positive residual indicates that the actual value (y) is greater than the predicted value ($\hat{y}$). Therefore, the student's actual final exam score was 5 points above what the model predicted based on their midterm score.

Question 8

1. The sum of the residuals is equal to zero. (correct answer)
2. The sum of the residuals is a positive value if the correlation is positive.
3. All residuals must be smaller in magnitude than the standard deviation of the y-values.
4. The sum of the absolute values of the residuals is equal to zero.

Explanation: A fundamental mathematical property of the least-squares regression line is that the sum of the residuals ($\sum(y_i - \hat{y}_i)$) is always zero.

Question 9

1. A linear model is not appropriate for this data; a nonlinear model would likely be a better fit. (correct answer)
2. The linear model is a good fit because the pattern is clear and easy to describe.
3. The variance of the residuals is not constant, which violates a condition for inference.
4. There is at least one influential outlier that is causing the curved pattern in the residuals.

Explanation: Any clear pattern, such as a curve, in a residual plot indicates that the relationship between the variables is not linear. Therefore, a linear model is not appropriate.

Question 10

1. 16.8 pounds (correct answer)
2. 18.0 pounds
3. 19.2 pounds
4. 15.6 pounds

Explanation: First, find the predicted weight: $\hat{y} = 5.4 + 2.1(6) = 18.0$ pounds. The residual is $y - \hat{y}$. So, $-1.2 = y - 18.0$. Solving for y gives the actual weight: $y = 18.0 - 1.2 = 16.8$ pounds.

Question 11

1. The point is an outlier because its actual y-value is far from the y-value predicted by the regression line. (correct answer)
2. The point is a high-leverage point because its x-value is far from the mean of the x-values.
3. The point must be an influential point that significantly changes the slope of the regression line.
4. The point's presence indicates that a linear model is inappropriate for the entire dataset.

Explanation: A large residual means the point lies far vertically from the regression line, which is the definition of an outlier in the context of regression. A point can be an outlier without being high-leverage or influential.

Question 12

1. 3.2 (correct answer)
2. 3.6
3. 4.0
4. 3.0

Explanation: The formula for a residual is $residual = actual \, y - predicted \, y$. Rearranging for the predicted value gives $predicted \, y = actual \, y - residual$. In this case, the predicted GPA is $3.4 - 0.2 = 3.2$.

Question 13

1. A linear model is not appropriate for describing the relationship in the original data. (correct answer)
2. The original data must have a strong positive linear association.
3. The sum of the residuals from the model will be a large positive number.
4. The y-intercept of the least-squares regression line must be positive.

Explanation: Any discernible pattern in a residual plot, including a linear pattern, indicates that the linear model does not adequately capture the relationship in the data. A random scatter is desired, so a linear model is not appropriate here.

Question 14

1. Nine residuals are positive and one residual is negative.
2. Five residuals are positive and five residuals are negative.
3. One residual is zero and the other nine residuals are positive. (correct answer)
4. Eight residuals are negative, one is positive, and one is zero.

Explanation: A key property of the least-squares regression line is that the sum of the residuals must be zero. It is impossible for one residual to be zero and the other nine to all be positive, as their sum would be greater than zero. The other options describe scenarios where the residuals could sum to zero.

Question 15

1. The residual for that point will increase in magnitude, and the line may shift, causing other residuals to change. (correct answer)
2. The residual for that point will increase in magnitude, but all other residuals will remain unchanged.
3. The sum of the residuals will no longer be zero because of the large change in the one point.
4. The sum of the squared residuals will decrease because the model will adjust to the new point.

Explanation: Moving a point vertically increases its distance from the original line, thus increasing the magnitude of its residual. Because the least-squares line is calculated based on all points, this change will cause the line itself to recalculate and shift (unless the point is influential in a very specific way), which in turn changes the predicted values and thus the residuals for other points.

Question 16

1. The assumption that the residuals are independent of each other. (correct answer)
2. The assumption that the relationship between sales and time is linear.
3. The assumption that the variance of the residuals is constant.
4. The assumption that the residuals are approximately normally distributed.

Explanation: When residuals are plotted against a time variable, a non-random pattern such as strings of positive or negative values suggests that the error term for one observation is related to the error term of a previous observation. This pattern indicates a lack of independence among the residuals (autocorrelation).

Question 17

1. The typical distance between an actual y-value and its predicted y-value is about 4.5 units. (correct answer)
2. The typical distance between an observed y-value and the mean of all the y-values is about 4.5 units.
3. The estimated slope of the least-squares regression line could be off by about 4.5 units from the true slope.
4. The predicted y-values are, on average, 4.5 units away from the mean of the observed y-values.

Explanation: The standard deviation of the residuals (s) measures the typical size of a prediction error (residual). It represents the approximate average vertical distance that the observed y-values fall from the fitted regression line.

Question 18

1. The model overestimated the selling price of this house by $12,000. (correct answer)
2. The model underestimated the selling price of this house by $12,000.
3. The actual selling price of the house was exactly $12,000.
4. The house has $12,000 less square footage than predicted for its price.

Explanation: A negative residual means the actual value is less than the predicted value ($y < \hat{y}$). This implies that the model's prediction was higher than the actual selling price, so the model overestimated the price.

Question 19

1. The residuals are plotted on the vertical axis, and the explanatory variable is plotted on the horizontal axis. (correct answer)
2. The response variable is plotted on the vertical axis, and the residuals are plotted on the horizontal axis.
3. The explanatory variable is plotted on the vertical axis, and the response variable is plotted on the horizontal axis.
4. The residuals are plotted on the vertical axis, and the response variable is plotted on the horizontal axis.

Explanation: A standard residual plot graphs the residuals ($y - \hat{y}$) on the vertical (y) axis against the explanatory variable (x) on the horizontal axis. Sometimes, predicted values ($\hat{y}$) are used on the horizontal axis instead.

Question 20

1. To assess the appropriateness of the linear model that was fitted to the data. (correct answer)
2. To determine the value of the correlation coefficient for the original data.
3. To directly calculate the slope and y-intercept of the least-squares regression line.
4. To visually display the strength and direction of the association between the two original variables.

Explanation: Residual plots are a key diagnostic tool used to check the assumptions of a linear model, such as the linearity of the relationship and the constant variance of errors, to determine if the model is appropriate.