Correlation Word Problems

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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
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) $r = 0.05$
3.
For a data set, $S_{xy} = 27$ , $S_{xx} = 20$ and $S_{yy} = 45$ . Compute $r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}$ .
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?
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 $r$ .
6.
A café finds $r = -0.72$ between the outdoor temperature and the number of hot chocolates sold per day. Interpret this value.
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 $1$
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
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 $S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y})$ .
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.
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.
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.
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.
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
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.
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.
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.
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.