Correlation Word Problems

Show your work for each problem.

1.
A survey of teenagers finds $r = -0.85$ between daily screen time and hours of sleep. Which description is correct?
1. a)Weak positive correlation
2. b)Very strong negative correlation
3. c)Moderate negative correlation
4. d)No correlation
Answer: Very strong negative correlation
$r$ is negative, so more screen time goes with less sleep. $|r| = 0.85$ lies between $0.80$ and $1.00$ , so the correlation is very strong.
2.
Four studies report these correlations. Which one shows the strongest linear relationship?
1. a) $r = 0.78$
2. b) $r = 0.45$
3. c) $r = -0.91$
4. d) $r = 0.05$
Answer: $r = -0.91$
The absolute values are $0.91$ , $0.45$ , $0.78$ and $0.05$ . The largest is $0.91$ , so $r = -0.91$ is the strongest (a very strong negative correlation).
3.
For a data set, $S_{xy} = 27$ , $S_{xx} = 20$ and $S_{yy} = 45$ . Compute $r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}$ .
Answer: 0.9
$r = \frac{27}{\sqrt{20 \cdot 45}} = \frac{27}{\sqrt{900}} = \frac{27}{30} = 0.9$ , a very strong positive correlation.
4.
Five students practised a song for 2, 3, 4, 5 and 6 hours and got these scores: $50$ , $55$ , $65$ , $70$ , $80$ . The first step to find $r$ is the means. What is $\bar{y}$ , the mean score?
Answer: 64
$\bar{y} = \frac{50 + 55 + 65 + 70 + 80}{5} = \frac{320}{5} = 64$ .
5.
A bakery records the number of hours it is open on five days: $2, 4, 6, 8, 10$ . Find the mean and the deviations needed for $r$ .
Answer: 0
1. What is the mean $\bar{x}$ ? 6
2. What is the deviation of $10$ , that is $10 - \bar{x}$ ? 4
3. What is the deviation of $2$ ? -4
4. What is the sum of all five deviations? 0
6.
A café finds $r = -0.72$ between the outdoor temperature and the number of hot chocolates sold per day. Interpret this value.
Answer: -0.72
1. What is $|r|$ ? 0.72
2. What is the direction? negative
3. How strong is the correlation? strong
4. As the temperature rises, sales of hot chocolate tend to... fall
7.
Data from many towns show $r = 0.90$ between the number of pharmacies and the number of hairdressers in a town. Which conclusion is justified?
1. a)Bigger towns have both more pharmacies and more hairdressers, so a causal claim is not justified
2. b)Opening a pharmacy attracts more hairdressers
3. c)A town can reduce its hairdressers by closing pharmacies
4. d) $r = 0.90$ proves causation because it is close to $1$
Answer: Bigger towns have both more pharmacies and more hairdressers, so a causal claim is not justified
The size of the town (its population) is a confounding variable: big towns have more people who need both pharmacies and hairdressers. The correlation is real, but it does not show that one causes the other, and no value of $r$ alone proves causation.
8.
A company wants to show that its new fertilizer causes faster plant growth. Which study could justify that causal claim?
1. a)A survey asking gardeners which fertilizer they use
2. b)Randomly assigning identical plants to get the fertilizer or not, with everything else kept the same
3. c)Computing $r$ for six farms that chose their own fertilizer
4. d)An online poll of customers
Answer: Randomly assigning identical plants to get the fertilizer or not, with everything else kept the same
Only a randomized controlled experiment can justify a causal claim. Random assignment spreads confounding variables (soil, sunlight, care) evenly over both groups. Farms that chose their own fertilizer may differ in many other ways.
9.
For five data points the deviations are $x_i - \bar{x}$ : $-2, -1, 0, 1, 2$ and $y_i - \bar{y}$ : $-4, -2, 1, 3, 2$ . Compute $S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y})$ .
Answer: 17
$8 + 2 + 0 + 3 + 4 = 17$ . The sum is positive, so $r$ will be positive.
10.
A study of car age and fuel efficiency gives $S_{xy} = -36$ , $S_{xx} = 40$ and $S_{yy} = 45$ . Compute $r$ and round to 2 decimal places.
Answer: -0.85
$r = \frac{-36}{\sqrt{1800}} \approx \frac{-36}{42.43} \approx -0.85$ . Older cars tend to be less fuel efficient (a very strong negative correlation).
11.
A football coach records training sessions and goals for five players: $(1, 2)$ , $(2, 6)$ , $(3, 4)$ , $(4, 5)$ , $(5, 8)$ . Compute $r$ to 2 decimal places.
Answer: 0.78
1. What is $\bar{x}$ ? 3
2. What is $\bar{y}$ ? 5
3. The $y$ deviations are $-3, 1, -1, 0, 3$ . What is $S_{xy}$ ? 11
4. What is $S_{xx}$ ? 10
5. What is $S_{yy}$ ? 20
6. What is $r$ , rounded to 2 decimal places? 0.78
12.
A teacher compares days absent with test scores (out of 10) for four students: $(1, 9)$ , $(3, 8)$ , $(5, 4)$ , $(7, 3)$ . Compute $r$ to 2 decimal places.
Answer: -0.96
1. What is $\bar{x}$ ? 4
2. What is $\bar{y}$ ? 6
3. The deviations are $x$ : $-3, -1, 1, 3$ and $y$ : $3, 2, -2, -3$ . What is $S_{xy}$ ? -22
4. What is $S_{xx}$ ? 20
5. What is $S_{yy}$ ? 26
6. What is $r$ , rounded to 2 decimal places? -0.96
13.
In a primary school with children aged 6 to 12, shoe size and reading score have $r = 0.88$ . A parent says: "Bigger feet make children read better." Judge this claim.
Answer: 0.88
1. What is the direction of the correlation? positive
2. How strong is it? very strong
3. Which confounding variable affects both shoe size and reading? age
4. Is the causal claim justified? no
14.
Five points lie close to a rising line and have $r = 0.95$ . A sixth point is added far to the right and far below the line. What happens to $r$ ?
1. a) $r$ gets much weaker and can even become negative
2. b) $r$ rises to exactly $1$
3. c) $r$ becomes larger than $1$
4. d) $r$ stays at $0.95$ , one point does not matter
Answer: $r$ gets much weaker and can even become negative
The new point has a large positive $x$ deviation and a large negative $y$ deviation, so it adds a big negative product to $S_{xy}$ . $r$ gets much weaker and can even turn negative, as in the lesson where one outlier changed $r$ from $1$ to $0$ . $r$ can never exceed $1$ .
15.
A shop sells four umbrella models and compares each price (in tens of dollars) with the number sold per week: $(2, 9)$ , $(4, 7)$ , $(6, 3)$ , $(8, 1)$ . Compute $r$ and round to 2 decimal places.
Answer: -0.99
$\bar{x} = 5$ , $\bar{y} = 5$ . Deviations: $x$ : $-3, -1, 1, 3$ ; $y$ : $4, 2, -2, -4$ . $S_{xy} = -12 - 2 - 2 - 12 = -28$ , $S_{xx} = 20$ , $S_{yy} = 40$ . $r = \frac{-28}{\sqrt{800}} \approx \frac{-28}{28.28} \approx -0.99$ : a very strong negative correlation.
16.
A basketball coach records extra practice hours and points scored for five players: $(1, 2)$ , $(2, 4)$ , $(3, 6)$ , $(4, 8)$ , $(10, 0)$ . The last player was injured early in the game. Compute $r$ with and without that player.
Answer: -0.45
1. What is $\bar{x}$ for all five players? 4
2. What is $\bar{y}$ for all five players? 4
3. The deviations are $x$ : $-3, -2, -1, 0, 6$ and $y$ : $-2, 0, 2, 4, -4$ . What is $S_{xy}$ ? -20
4. What is $S_{xx}$ ? 50
5. What is $S_{yy}$ ? 40
6. What is $r$ with all five players, rounded to 2 decimal places? -0.45
7. The other four points lie on $y = 2x$ . What is $r$ without the injured player? 1
17.
A city planner compares the distance from the city center (km) with the number of cafés within walking distance: $(2, 8)$ , $(4, 10)$ , $(6, 5)$ , $(8, 5)$ . Compute $r$ and round to 2 decimal places.
Answer: -0.74
1. What is $\bar{x}$ ? 5
2. What is $\bar{y}$ ? 7
3. The deviations are $x$ : $-3, -1, 1, 3$ and $y$ : $1, 3, -2, -2$ . What is $S_{xy}$ ? -14
4. What is $S_{xx}$ ? 20
5. What is $S_{yy}$ ? 18
6. What is $r$ , rounded to 2 decimal places? -0.74
18.
For study hours and test scores of five students, $S_{xy} = 45$ , $S_{xx} = 10$ and $S_{yy} = 250$ . Compute $r$ , then see what happens when the data are changed.
Answer: 0.9
1. What is $\sqrt{S_{xx} \cdot S_{yy}}$ ? 50
2. What is $r$ ? 0.9
3. The teacher adds 5 bonus points to every score. What is the new $r$ ? 0.9
4. Study time is now recorded in minutes instead of hours. What is the new $r$ ? 0.9