Interpreting Correlation

Show your work for each problem.

1.
On a mountain hike, the correlation between altitude and air temperature is $r = -0.92$ . How do you describe it?
1. a)Weak negative
2. b)Strong positive
3. c)Strong negative
4. d)No linear correlation
Answer: Strong negative
$r$ is negative, so the direction is negative. $|-0.92| = 0.92 \ge 0.7$ , so it is strong: the higher you climb, the colder it tends to be.
2.
Which value of $r$ shows the weakest linear relationship?
1. a) $0.47$
2. b) $-0.08$
3. c) $0.56$
4. d) $-0.85$
Answer: $-0.08$
The absolute values are $0.85$ , $0.47$ , $0.08$ and $0.56$ . The smallest is $0.08$ , so $r = -0.08$ is the weakest (almost no linear relationship).
3.
The correlation between the number of hours a phone is used per day and its remaining battery in the evening is $r = -0.54$ . What is $|r|$ , the number that gives the strength?
Answer: 0.54
$|-0.54| = 0.54$ . Since $0.4 \le 0.54 < 0.7$ , this is a moderate negative correlation.
4.
Four studies report $r = 0.35$ , $r = -0.81$ , $r = 0.57$ and $r = -0.18$ . Type the value of $r$ that shows the strongest linear relationship.
Answer: -0.81
The absolute values are $0.35$ , $0.81$ , $0.57$ and $0.18$ . The largest is $0.81$ , so $r = -0.81$ is the strongest (a strong negative correlation).
5.
For 30 students, the correlation between study time per week and the grade on a final exam is $r = 0.78$ . Describe the correlation step by step.
Answer: 0.78
1. Is $r$ positive or negative? positive
2. What is $|r|$ ? 0.78
3. Is the correlation strong, moderate or weak? strong
6.
In a survey of 80 families, the correlation between the distance from home to school (km) and the number of days per week the children walk to school is $r = -0.47$ . Find the strength step by step.
Answer: 0.47
1. What is $|r|$ ? 0.47
2. Is $|r| \ge 0.7$ ? Type 1 for yes, 0 for no. 0
3. Is $|r| \ge 0.4$ ? Type 1 for yes, 0 for no. 1
4. So the correlation is: moderate
7.
For 50 teenagers, the correlation between hours of sleep and the score on a reaction test (higher is better) is $r = 0.52$ . Which sentence is the best interpretation?
1. a)52% of the teenagers who sleep more score higher.
2. b)There is a moderate negative correlation: teenagers who sleep more tend to score lower.
3. c)There is a moderate positive linear correlation: teenagers who sleep more tend to score higher.
4. d)There is a strong positive correlation, so more sleep causes higher scores.
Answer: There is a moderate positive linear correlation: teenagers who sleep more tend to score higher.
$0.4 \le 0.52 < 0.7$ and $r > 0$ : moderate positive. "Strong" and "causes" claim too much, "52%" misreads $r$ as a percentage, and "negative" has the wrong direction.
8.
A scatter plot shows a clear U shape, but the calculator gives $r \approx 0.01$ . Which conclusion is correct?
1. a)The value of $r$ must have been calculated wrongly
2. b)There is almost no linear relationship, but a strong curved one
3. c)There is no relationship at all between the variables
4. d)There is a strong positive linear correlation
Answer: There is almost no linear relationship, but a strong curved one
$r$ only measures straight-line relationships. A U shape goes down and then up, so the two parts cancel and $r$ is close to $0$ , even though the curved relationship is strong.
9.
For 25 adults, the correlation between height in cm and weight in kg is $r = 0.71$ . A researcher changes the heights to meters and the weights to grams. What is $r$ now?
Answer: 0.71
Heights are divided by $100$ and weights are multiplied by $1000$ : both are positive numbers, so the pattern stays the same. $r$ has no units and stays $0.71$ .
10.
Using the rule of thumb ( $|r| \ge 0.7$ is strong), how many of these values show a strong correlation? $0.72$ , $-0.91$ , $0.59$ , $-0.45$ , $0.70$ , $0.38$
Answer: 3
Strong: $0.72$ , $-0.91$ (because $|-0.91| = 0.91$ ) and $0.70$ . Not strong: $0.59$ , $-0.45$ , $0.38$ . That makes $3$ .
11.
Study A finds $r = -0.74$ between the price of a concert ticket and the number of tickets sold. Study B finds $r = 0.56$ between advertising spending and tickets sold. Which correlation is stronger?
Answer: -0.74
1. What is $|r|$ for study A? 0.74
2. What is $|r|$ for study B? 0.56
3. Type the value of $r$ (with its sign) of the stronger correlation. -0.74
12.
For 40 taxi rides, the correlation between distance in km and fare in dollars is $r = 0.83$ . The company then writes every fare in cents and every distance in miles.
Answer: 0.83
1. To write a fare in cents, you multiply it by which number? 100
2. What is $r$ after the fares are written in cents? 0.83
3. 1 mile is about 1.609 km, so each distance is divided by 1.609. What is $r$ after both changes? 0.83
13.
For 60 students, the correlation between hours of video games per week and the grade average (higher is better) is $r = -0.58$ . Build an interpretation sentence.
Answer: -0.58
1. Direction of the correlation: negative
2. Strength of the correlation: moderate
3. Which ending fits an interpretation sentence? students who game more tend to have lower grades
14.
Eight points form a round cloud with no pattern, $r \approx 0.05$ . One new point is added far away in the top right corner of the scatter plot. What happens to $r$ ?
1. a)It becomes strongly negative
2. b)It becomes strongly positive
3. c)It stays about $0.05$
4. d)It becomes exactly $0$
Answer: It becomes strongly positive
The new point is far above and to the right of all others, so it pulls the pattern into a rising line: $r$ jumps to a strong positive value. One outlier can create a strong correlation that the other data do not support.
15.
The points $(-2, 4)$ , $(-1, 1)$ , $(0, 0)$ , $(1, 1)$ , $(2, 4)$ lie exactly on the curve $y = x^2$ . Here $\bar{x} = 0$ and $\bar{y} = 2$ , and the numerator of $r$ is $\sum (x - \bar{x})(y - \bar{y})$ . What is $r$ ?
Answer: 0
The products are $(-2)(2) = -4$ , $(-1)(-1) = 1$ , $0 \cdot (-2) = 0$ , $1 \cdot (-1) = -1$ , $2 \cdot 2 = 4$ . Their sum is $0$ , so $r = 0$ , although every point lies exactly on a curve: $r$ only measures linear relationships.
16.
A ball is thrown up. Its height (m) after $0, 1, 2, 3, 4$ seconds is $0, 3, 4, 3, 0$ . Find $r$ between time and height and interpret it.
Answer: 0
1. What is the mean time $\bar{x}$ ? 2
2. What is the mean height $\bar{y}$ ? 2
3. Compute $\sum (x - \bar{x})(y - \bar{y})$ . 0
4. So what is $r$ ? 0
5. What does the scatter plot show? a curved relationship
17.
Plot E: six students give $r \approx 0.999$ between practice hours and quiz score. With the seventh student, the point $(7, 40)$ , $r \approx 0.14$ . Analyze the outlier.
Answer: 1
1. Strength without the outlier ( $r \approx 0.999$ ): strong
2. Strength with the outlier ( $r \approx 0.14$ ): weak
3. How many points caused the change? 1
4. The score 40 was a typing error: the real score was 90. The point $(7, 90)$ continues the pattern. Is the new $r$ closer to $0$ or to $1$ ? Type that number. 1
18.
An ice cream shop finds $r = 0.9$ between the daily temperature in °C and the number of ice creams sold. The owner converts the temperatures to °F with $F = 1.8C + 32$ and then also swaps the axes.
Answer: 0.9
1. Convert $25\,°C$ to °F. 77
2. What is $r$ between the temperature in °F and ice creams sold? 0.9
3. Now ice creams sold are on the $x$ -axis and temperature on the $y$ -axis. What is $r$ ? 0.9