Correlation Coefficient

Show your work for each problem.

1.
What does a correlation coefficient of indicate?
1. a)Perfect negative correlation
2. b)Perfect positive correlation
3. c)Moderate correlation
4. d)No correlation
Answer: Perfect positive correlation
indicates a perfect positive correlation, meaning all data points lie exactly on an upward-sloping line. As one variable increases, the other increases proportionally.
2.
Which value of indicates the strongest correlation?
1. a)
2. b)
3. c)
4. d)
Answer:
has , which is the largest absolute value. The negative sign indicates direction (inverse relationship), not weakness. Compare: .
3.
If , what is ?
Answer: 0.72
. The absolute value is simply the number without its sign, representing the strength of the correlation.
4.
A study finds that ice cream sales and drowning incidents both increase in summer. What can we conclude?
1. a)They are correlated, but ice cream doesn't cause drowning
2. b)Drowning causes people to buy ice cream
3. c)There is no relationship
4. d)Ice cream causes drowning
Answer: They are correlated, but ice cream doesn't cause drowning
This is a classic example of correlation without causation. Hot weather (the confounding variable) causes both increased ice cream sales and more swimming, which leads to more drownings. Ice cream itself doesn't cause drowning.
5.
A correlation of means what percentage of variance is explained? (Use )
Answer: 49
. This means 49% of the variance in one variable is explained by the other. The remaining 51% is due to other factors.
6.
How would you describe a correlation of ?
1. a)Strong positive correlation
2. b)Weak positive correlation
3. c)Moderate positive correlation
4. d)Very strong positive correlation
Answer: Moderate positive correlation
With , this falls in the moderate range (0.40-0.59). The positive sign indicates that as one variable increases, the other tends to increase as well.
7.
Given , , , , . Calculate .
Answer: 1
1. What is the numerator of the formula? 40
2. What is the product in the denominator: ? 1600
3. What is ? 40
4. What is ? 1
8.
If the product , what is the denominator of the formula?
Answer: 15
The denominator is .
9.
For the data points : Calculate the means and then find .
Answer: 1
1. What is ? 4
2. What is ? 7
3. Notice that y = 2x - 1 for all points. What type of relationship is this? linear
4. For a perfect linear relationship with positive slope, what is ? 1
10.
A researcher adds an outlier to their data and changes from to . What does this demonstrate?
1. a)Outliers don't affect
2. b) is sensitive to outliers
3. c) is always reliable
4. d)The data has no correlation
Answer: is sensitive to outliers
This demonstrates that is sensitive to outliers. A single extreme point can dramatically change the correlation coefficient. This is why it's important to always visualize data with a scatter plot before relying on .
11.
Data: . Calculate the correlation coefficient step by step.
Answer: 0.8
1. Calculate 2.5
2. Calculate 2.5
3. Calculate . Deviations: . Products sum to? 4
4. Calculate 5
5. By symmetry, also equals? 5
6. Calculate 0.8
12.
A scatter plot shows points forming a perfect U-shape. What is the likely value of ?
1. a)Close to -1
2. b)Close to 0
3. c)Exactly 0.5
4. d)Close to 1
Answer: Close to 0
A U-shape is a non-linear (quadratic) pattern. Since only measures linear correlation, the positive and negative slopes on either side of the U cancel out, resulting in . This demonstrates a key limitation: doesn't mean no relationship exists!
13.
If , what percentage of variance in is NOT explained by ?
Answer: 19
explained. Therefore, is NOT explained by the linear relationship with .
14.
Compare two datasets: Dataset A has , Dataset B has . Which has stronger predictive power?
Answer: B
1. What is for Dataset A? 0.65
2. What is for Dataset B? 0.78
3. Which absolute value is larger? 0.78
4. Which dataset has stronger correlation (A or B)? B
15.
What can you conclude if between two variables?
1. a)The data has errors
2. b)No relationship of any kind exists
3. c)No linear relationship exists
4. d)A perfect relationship exists
Answer: No linear relationship exists
means there is no LINEAR relationship. However, there could still be a non-linear relationship (quadratic, exponential, etc.). Always check the scatter plot to see if patterns exist that cannot detect.
16.
Two stocks have . If Stock A drops 10%, approximately how much would you expect Stock B to change (in %)?
Answer: -10
With (very strong positive correlation), when Stock A drops 10%, Stock B would be expected to drop approximately 10% as well. The high correlation means they move together in the same direction.