Let's go over the questions based on the image:

Section 2: Scatterplot Analysis

(a) Indicate on the graph which line is which

• There are two lines: one with the outlier included and one without. The one labeled "Outlier Included" should be the line that doesn't fit the overall trend as well (likely less steep or further away from most of the points). The "Outlier Omitted" line should have a better fit, with a steeper slope that better represents the central cluster of data points.

(b) Estimate the Pearson correlation for each

• For the line with the outlier included: Pearson correlation might be lower due to the outlier's influence. The estimate given on the image is around -0.51.
• For the line with the outlier omitted: The estimate would likely be higher, as removing the outlier results in a stronger negative linear trend. The estimated Pearson correlation is -0.75 (as noted in the image).

(c) Which line has the steeper slope?

• The line without the outlier will have a steeper slope because the outlier drags the slope down when included.

(d) Which line has the greater y-intercept?

• The line with the outlier should have a greater y-intercept, as the outlier pulls the line upward.

Section 3: Statistical Graph (Ogive) Analysis

(a) Give the report for this scatterplot

• The scatterplot is noted as having a "very strong negative" relationship.

(b) Estimate the Pearson correlation

• Since it has a strong negative relationship, the Pearson correlation should be close to -1.

(c) Estimate the Spearman rank correlation

• Since Spearman's rank correlation also measures monotonic relationships, it should be close to -1 as well, reflecting a strong negative rank correlation.

(d) Which correlation is best suited for this situation: Pearson or Spearman? Why?

• Spearman correlation might be more appropriate if the relationship is not linear but still monotonic (for example, if there's a non-linear but consistent downward trend).

Section 4: Height and Shoe Size Correlation

(a) Rank both lists

• The ranks for height and corresponding shoe sizes are already given in the table beneath. The values for heights are ranked as:
  Heights (in cm): 8, 9, 6, 8, 7, 10, 2.5, 5, 11, 9
  Shoe sizes: 6, 5, 8, 7, 10, 5, 1, 2.5, 9, 5

Correlation Change in Different Scenarios

(b) How would the correlation change if height was converted from cm to inches?

• No change in correlation because Pearson and Spearman correlations are scale-invariant (they don’t depend on the units of measurement).

(c) How would the correlation change if shoe size was converted from US to UK?

• No change in correlation since correlation only reflects the relationship between ranks or values, not their specific unit of measure.

(d) How would the correlation change if we switched the explanatory and response variables?

• The Pearson correlation value would remain the same, as correlation does not depend on which variable is explanatory or response.

Let me know if you'd like more details on any part of this, or if you have any specific questions!

5 Additional Questions:

1. How is Pearson correlation calculated, and why is it sensitive to outliers?
2. What are the key differences between Pearson and Spearman correlations?
3. How would removing more outliers affect both the slope and correlation in the scatterplot?
4. Can a correlation coefficient of -1 or 1 always imply causation?
5. How do ranks affect the Spearman correlation when there are tied values?

Tip:

Remember that outliers can significantly affect the Pearson correlation but usually have less impact on Spearman, as it’s based on ranks!