Bluebook SAT Test Problem Solving and Data Analysis 4

Question 1

In a linear regression model, the residual for a data point is defined as the observed value minus the predicted value. A data scientist notes that a particular point has an observed y-value of 23 and a residual of −7. What is the predicted value of y for this data point according to the regression model?

Accepted Answer

30

Answer

The residual formula is: residual = observed − predicted. Substituting the known values: −7 = 23 − predicted. Solving: predicted = 23 − (−7) = 30. A negative residual means the model overpredicted the actual value, so the predicted value (30) is higher than the observed value (23).

Question 2

In a chemistry class, 24 students who completed the required lab earned an average score of 78. The remaining 6 students who did not complete the lab earned an average score of 48. What is the overall class average score?

Accepted Answer

72

Answer

The overall average requires weighting each group by its size. Total points from lab completers: 24 × 78 = 1,872. Total points from non-completers: 6 × 48 = 288. Combined total: 1,872 + 288 = 2,160. Total students: 24 + 6 = 30. Overall average: 2,160 ÷ 30 = 72. Simply averaging 78 and 48 gives 63, which is incorrect because the groups are different sizes.

Question 3

A research study surveyed 400 adults about exercise and health outcomes. Of the 240 adults who exercise regularly, 180 reported good health and 60 reported poor health. Of the 160 adults who do not exercise, 40 reported good health and 120 reported poor health. Given that a randomly selected adult from this study reported good health, what is the probability that they exercise regularly?

Accepted Answer

9/11

Answer

This is a conditional probability problem. We want P(exercises | good health). The total number of adults with good health is 180 + 40 = 220. Of those, 180 exercise regularly. Therefore, P(exercises | good health) = 180/220 = 9/11. This is NOT the same as P(good health | exercises) = 180/240 = 3/4, which is a common error involving the reversal of the condition.

Question 4

A retail company's revenue increased by 40% in Year 1, then decreased by 25% in Year 2, and decreased by an additional 20% in Year 3. What is the overall percent change in revenue from the beginning of Year 1 to the end of Year 3?

Accepted Answer

−16%

Answer

Let the initial revenue equal R. After Year 1 (40% increase): R × 1.40 = 1.40R. After Year 2 (25% decrease): 1.40R × 0.75 = 1.05R. After Year 3 (20% decrease): 1.05R × 0.80 = 0.84R. The overall change is 0.84R − R = −0.16R, which is a 16% decrease. Each percent change must be applied sequentially to the result of the prior year, not to the original value.

Question 5

A data set contains 10 values with a mean of 40 and a standard deviation of 6. A new value of 40 is added to the data set, creating a set of 11 values. Which of the following must be true about the new data set?

Accepted Answer

The mean stays the same and the standard deviation decreases

Answer

Adding the value 40, which equals the mean, does not change the mean — it remains 40. However, the standard deviation does change. The new value contributes a squared deviation of (40 − 40)² = 0, while the variance is now spread over 11 values instead of 10. New variance = (10 × 36 + 0) / 11 = 360/11 ≈ 32.7, giving a new standard deviation of approximately 5.72, which is less than the original 6. Adding the mean to a data set always decreases the standard deviation because it pulls the spread inward without adding any new dispersion.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Problem Solving and Data Analysis 4