Problem Solving and Data Analysis Flashcards

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?

Answer: 30

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).

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?

Answer: 72

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.

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?

Answer: 9/11

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.

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?

Answer: −16%

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.

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?

Answer: The mean stays the same and the standard deviation decreases

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.

A polling organization surveyed 800 randomly selected residents of a city. The poll found that 62% of respondents support a new transit initiative, with a margin of error of ±4 percentage points at a 95% confidence level. A second, independent poll of 3,200 residents found 60% support. Which of the following conclusions is most statistically sound when comparing the two polls?

Answer: The second poll is more reliable because its larger sample size produces a smaller margin of error

Margin of error decreases as sample size increases — specifically, it is inversely proportional to the square root of n. The second poll's sample is 4× larger, so its margin of error is roughly half as large (±2%), making it more precise. The two polls are not contradictory: both 62% and 60% fall within each other's confidence intervals, suggesting consistency. Option A confuses point estimates with intervals. Option C mischaracterizes how margins of error work — they quantify uncertainty, not unreliability. Option D misdefines confidence level, which applies to the interval construction method, not the probability of a single value.