1. Which scatterplot(s) would fitting a linear regression model be appropriate? (Select all that apply.)

• ✅ a

• ❌ b

• ✅ c

• ✅ d

• ❌ e

Explanation:
Linear regression is appropriate when the cloud of points shows an approximately straight-line pattern (positive or negative). I picked a, c, and d as those exhibiting reasonably linear patterns; b and e look non-linear or random.

---

2. Which scatterplot(s) would have a correlation coefficient close to 0? (Select all that apply.)

• ❌ a

• ✅ b

• ❌ c

• ❌ d

• ✅ e

Explanation:
Correlation near 0 occurs when there is little-to-no linear association. Plots b and e were chosen as showing no clear linear trend (random / circular patterns).

---

3. Which scatterplot has the highest absolute correlation (strongest linear relationship)?

• ❌ a

• ✅ b

• ❌ c

• ❌ d

Explanation:
The plot chosen shows the tightest straight-line pattern (points closely clustered around a line), indicating the largest absolute correlation.

---

4. What distribution do the true errors need to follow to perform inference procedures in linear regression?

• ❌ True errors must be N(0,1)

• ✅ True errors must be N(0, σ²)

• ❌ True errors must be Uniformly distributed

• ❌ True errors do not need any specific distribution

Explanation:
For standard t-tests and confidence intervals in linear regression we assume errors are normally distributed with mean 0 and constant variance σ². (Note: large-sample inference can be robust, but the classical assumption is Normal(0, σ²).)

---

5. Which are assumptions needed for hypothesis test on population slope? (Select all that apply.)

• ✅ True errors must be normally distributed.

• ✅ True errors have constant variance.

• ✅ The population relationship between dependent and explanatory variable is linear.

Explanation:
All three are part of the classical linear regression assumptions needed for valid inference on the slope (normality of errors, homoscedasticity, and correct linear form).

---

6. Interpretation of estimated slope b1 = 0.21 (predicting hotel rating from nightly cost)

• ❌ When a hotel’s nightly cost is $0 the hotel’s rating is expected to be 0.21 points.

• ❌ When a hotel rating is 0 points the hotel’s nightly cost is expected to be $0.21 dollars.

• ✅ The hotel rating is estimated to increase by 0.21 points for every additional dollar spent on nightly hotel cost, on average.

• ❌ The nightly hotel cost is estimated to increase by $0.21 dollars for every additional hotel rating point, on average.

Explanation:
The slope is change in response (rating) per unit change in predictor (dollars).

---

7. Residual for subject with head size 3500 cm³ and brain weight 1430.86 grams

• ✅ -183.3 grams

• ❌ 183.3 grams

• ❌ -4195.752 cm³

• ❌ 4195.752 cm³

Explanation:
Residual = observed − predicted. The numeric answer corresponds to the observed minus the model’s predicted brain weight for head size 3500 (gives −183.3 g).

---

8. Interpretation of R² = 0.6393

• ❌ 0.6393% of the variation …

• ✅ 63.93% of the variation in brain weight can be accounted for by the linear relationship with head size.

• ❌ We would expect brain weight to increase by 0.6393 grams for every additional cm³ …

• ❌ We would expect head size to increase by 0.6393 cm³ for every additional gram …

Explanation:
R² (0.6393) means ~63.93% of the variance in the response is explained by the predictor.

---

9. Appropriate p-value for testing a significant positive linear relationship

• ✅ 4.61e-11

• ❌ <2e-16

• ❌ 2.305e-11

• ❌ <1e-16

Explanation:
The reported p-value from the regression output is 4.61 × 10⁻¹¹, indicating a very small p-value (strong evidence of a relationship).

---

10. How does the 95% prediction interval width compare to the 95% confidence interval for the mean (both at head size 3400 cm³)?

• ✅ Wider

• ❌ Narrower

• ❌ Stays the same

Explanation:
Prediction intervals for an individual observation are wider than confidence intervals for the mean because they include both uncertainty in the mean and the residual variability.

---

11. How would the 95% confidence interval width at 3600 cm³ compare to that at 3400 cm³?

• ✅ Wider

• ❌ Narrower

• ❌ Stay the same

Explanation:
If 3600 cm³ is farther from the mean head size than 3400 cm³, the CI for the mean at 3600 will be wider (variance of the fitted mean increases as you move away from the mean of the x-values).

---

12. Cautions predicting brain weight for an 8-year-old with head size 1800 cm³ (select all that apply)

• ❌ Correlation does not imply causation for brain weight.

• ✅ Extrapolation — A head size of 1800 cm³ is outside the range of our data.

• ✅ Extrapolation — The model was created using only data for adults, not children.

• ❌ We do not know if the child is male or female.

• ❌ No cautions needed

Explanation:
The main issues are extrapolation (value outside observed adult range) and the model being built on adults only; predictions for a child are unreliable. (While causation is a general caveat, the two extrapolation cautions are the most directly relevant.)

---

13. Interpretation of estimated coefficient for age = −23.97

• ❌ The average brain weight for younger subjects is estimated to be 23.97 grams less than the average brain weight for older subjects.

• ❌ Keeping head size and sex constant, the average brain weight for younger subjects is estimated to be 23.97 grams less than the average brain weight for older subjects.

• ❌ The average brain weight for older subjects is estimated to be 23.97 grams less than the average brain weight for younger subjects.

• ✅ Keeping head size and sex constant, the average brain weight for older subjects is estimated to be 23.97 grams less than the average brain weight for younger adults.

Explanation:
Age is coded 0=young, 1=old; a negative coefficient for age means older subjects (age=1) have an average brain weight ~23.97 g less than younger subjects (age=0), holding head size and sex constant.

---

🧾 Summary Table

---