Introduction to Correlation

Show your work for each problem.

1.
When both variables increase together, this is called:
1. a)Causation
2. b)Positive correlation
3. c)Negative correlation
4. d)No correlation
Answer: Positive correlation
When both variables increase together (or both decrease together), this is positive correlation. They move in the same direction.
2.
The correlation coefficient $r$ always falls between which values?
1. a) $0$ and $1$
2. b) $-1$ and $1$
3. c) $-100$ and $100$
4. d) $0$ and $100$
Answer: $-1$ and $1$
The correlation coefficient $r$ is always between $-1$ and $1$ . A value of $-1$ means perfect negative correlation, $0$ means no correlation, and $1$ means perfect positive correlation.
3.
Which value of $r$ means there is no linear correlation at all between two variables? Type the number.
Answer: 0
$r = 0$ means no correlation (no linear relationship) between the variables. Values close to $0$ , like $0.05$ , also mean almost no correlation.
4.
As the price of a used car decreases with age, this represents:
1. a)No correlation
2. b)Negative correlation
3. c)Perfect correlation
4. d)Positive correlation
Answer: Negative correlation
This is negative correlation. As age increases, price decreases. The variables move in opposite directions.
5.
Which value of $r$ means a perfect negative correlation, where all points lie exactly on a falling straight line? Type the number.
Answer: -1
$r = -1$ is a perfect negative correlation: all points lie exactly on a straight line that falls from left to right. It is the smallest value $r$ can have.
6.
Which correlation coefficient represents the STRONGEST relationship?
1. a) $r = -0.92$
2. b) $r = 0.30$
3. c) $r = 0.75$
4. d) $r = -0.50$
Answer: $r = -0.92$
$r = -0.92$ represents the strongest relationship. Although it's negative, $|-0.92| = 0.92$ is closest to $1$ . The sign indicates direction (negative), while the magnitude indicates strength.
7.
Classify the correlation $r = -0.78$ by its direction and strength.
Answer: strong negative
1. Is the correlation positive or negative? negative
2. What is the absolute value $|{-}0.78|$ ? 0.78
3. Is 0.78 closer to 0, 0.5, or 1? 1
4. What strength does this indicate? (weak, moderate, or strong) strong
8.
A study finds $r = 0.52$ . How strong is this correlation?
1. a)Moderate
2. b)Strong
3. c)Weak
4. d)Perfect
Answer: Moderate
A correlation of $r = 0.52$ is moderate. Values of $\lvert r \rvert$ from 0.4 to just under 0.7 (so also $r = -0.52$ ) show a moderate relationship: neither weak nor strong.
9.
Ice cream sales and drowning incidents both increase in summer. This is an example of:
1. a)Correlation without causation
2. b)Negative correlation
3. c)No correlation
4. d)Causation
Answer: Correlation without causation
This is correlation without causation. Both variables are correlated (they increase together), but ice cream doesn't cause drowning. A third variable (hot summer weather) causes both: more people buy ice cream AND more people go swimming.
10.
A researcher finds that as study time increases, test anxiety decreases. Describe this correlation.
Answer: negative correlation
1. What happens to study time? increases
2. What happens to test anxiety? decreases
3. Do the variables move in the same direction or opposite directions? opposite
4. What type of correlation is this? negative
11.
A scatter plot shows points going upward from left to right. The calculated correlation is $r = 0.83$ . Interpret this fully.
Answer: strong positive correlation
1. Based on the visual (upward trend), is this positive or negative? positive
2. Does the $r$ value confirm this? yes
3. How close is 0.83 to 1? very close
4. What is the strength of this correlation? strong
5. Choose the complete interpretation: direction + strength strong positive
12.
Which statement correctly describes the relationship between correlation and causation?
1. a)Correlation always implies causation
2. b)Correlation and causation are the same thing
3. c)Correlation can exist without causation
4. d)Causation cannot exist with correlation
Answer: Correlation can exist without causation
Correlation can exist without causation. Two variables can be related without one causing the other. The ice cream/drowning example shows this: both increase in summer due to a third variable (hot weather), but ice cream doesn't cause drowning.
13.
Research shows that countries with more Nobel Prize winners also have higher chocolate consumption. The correlation is $r = 0.79$ . Does this prove that chocolate makes people smarter?
1. a)No: $r = 0.79$ is too weak to show any relationship
2. b)Yes: the correlation is positive
3. c)No: a third factor, such as a country's wealth, may explain both
4. d)Yes: $r = 0.79$ is a strong correlation
Answer: No: a third factor, such as a country's wealth, may explain both
No, we cannot conclude that chocolate makes people smarter. This is correlation, not causation. A third variable might explain both: wealthy countries can afford both more education (leading to Nobel Prizes) AND more luxury goods (like chocolate). Correlation ≠ Causation!
14.
Order these correlations from weakest to strongest relationship: $r = 0.3$ , $r = -0.85$ , $r = 0.65$ , $r = -0.15$ . Which value of $r$ comes third?
Answer: 0.65
1. What is $|-0.15|$ , the strength of $r = -0.15$ ? 0.15
2. Which value of $r$ is the weakest? -0.15
3. Which value of $r$ is the strongest? -0.85
4. Which value of $r$ comes second? 0.3
5. Which value of $r$ comes third? 0.65
15.
A study finds that students who eat breakfast score 10% higher on tests. The correlation is $r = 0.45$ . What can we conclude?
1. a)Students should skip breakfast to focus on studying
2. b)Eating breakfast causes higher test scores
3. c)There is a moderate positive correlation between breakfast and scores
4. d)There is no relationship between breakfast and test scores
Answer: There is a moderate positive correlation between breakfast and scores
We can only conclude there is a moderate positive correlation ( $r = 0.45$ is between 0.4 and 0.7). We CANNOT say breakfast causes better scores - that would be concluding causation from correlation. Maybe students who eat breakfast also have other healthy habits, more stable home lives, or better sleep - we don't know the cause.