Unit Quiz: Advanced Data Analysis

1. r-squared = (0.72)^2 = 0.5184 or about 51.84%. This means approximately 52% of the variation in cardiovascular health scores can be explained by the linear relationship with hours of exercise per week. The remaining 48% is due to other factors.

2. Predicted score = 45 + 6(7) = 45 + 42 = 87. Residual = Actual - Predicted = 82 - 87 = -5. The negative residual indicates the student scored 5 points below what the model predicted for someone who studied 7 hours.

3. Confidence interval: 48% plus or minus 3.2% = (44.8%, 51.2%). Since the interval includes values below 50%, the data does NOT support the claim that Candidate A has majority support. The true proportion could be anywhere from about 45% to 51%.

4. B) r = -0.89. Strength is determined by the absolute value. |-0.89| = 0.89 is the largest absolute value among the choices, indicating the strongest relationship (even though it is negative).

5. This is extrapolation beyond the range of the data. The model was built using x values from 20 to 80, so predicting at x = 150 is unreliable. The linear relationship may not hold that far outside the observed range, and the prediction could be very inaccurate.

6. B) There is a strong positive association between population and number of hospitals. This is the only answer that correctly describes correlation without implying causation. Options A and C incorrectly claim causation, and D misinterprets the correlation value.

7. Since advertising is in thousands, use x = 25: Revenue = 120 + 8.5(25) = 120 + 212.5 = 332.5. The predicted revenue is $332,500 (since revenue is also in thousands).

8. Poll A: (48%, 56%), Poll B: (44%, 50%), Poll C: (47%, 51%). Looking for overlap: Poll A's lower bound (48%) and Poll C's interval (47%-51%) overlap between 48% and 51%. Poll B (44%-50%) overlaps with Poll C between 47% and 50%. All three have some overlap in the range of approximately 48% to 50%. This suggests the polls are reasonably consistent with each other.

9. The error is confusing r and r-squared. If r-squared = 0.81, then r = plus or minus the square root of 0.81 = plus or minus 0.9. The correlation is 0.9 (or -0.9), not 0.81. Without additional context, we cannot determine the sign.

10. The journalist is making a correlation-causation error. The lurking variable is fire size/severity. Larger fires require more firefighters AND cause more damage. The firefighters are responding to damage that is already occurring or inevitable; they do not cause the damage. This is a classic example of confounding.

11. Slope: For each one-unit increase in shoe size, predicted height increases by 1.8 inches. Y-intercept: When shoe size is 0, predicted height is 50 inches. The y-intercept is NOT meaningful in this context because a shoe size of 0 is outside any realistic range and the relationship would not extend that far.

12.
a) Sample proportion = (0.38 + 0.46) / 2 = 0.42 or 42%
b) Margin of error = (0.46 - 0.38) / 2 = 0.04 or 4%
c) The claim is refuted. The entire confidence interval (38% to 46%) is below 50%, so we are 95% confident that the true proportion of adults who exercise regularly is less than half. The company's claim is not supported by this data.