Correlation vs Causation

Show your work for each problem.

1.
In a beach town, ice cream sales and sunburn cases have a very strong positive correlation. Which explanation is most likely?
1. a)Getting sunburned makes people buy ice cream
2. b)Eating ice cream causes sunburn
3. c)The data must be wrong
4. d)Hot, sunny weather increases both
Answer: Hot, sunny weather increases both
Hot, sunny weather is a confounding variable: it brings more people to the beach, who buy more ice cream and get more sunburns. Neither variable causes the other.
2.
In a randomized experiment, $14$ of the $200$ people in the treatment group got sick. What percent of the treatment group got sick? (Enter the number only.)
Answer: 7
$\frac{14}{200} \times 100 = 7$ , so $7\%$ of the treatment group got sick.
3.
To test a new fertilizer, $100$ plants are split at random into two groups of $50$ . In the fertilizer group, $40$ plants grow taller than $30$ cm; in the control group, $25$ plants do. By how many percentage points is the rate higher in the fertilizer group?
Answer: 30
1. What percent of the fertilizer group grew taller than $30$ cm? 80
2. What percent of the control group grew taller than $30$ cm? 50
3. What is the difference in percentage points? 30
4.
Which kind of study can show that one variable causes another?
1. a)A scatter plot with $r = 0.9$
2. b)A survey of $1000$ people
3. c)Data collected over past years
4. d)A randomized controlled experiment
Answer: A randomized controlled experiment
Only a randomized controlled experiment assigns the treatment by chance, so the groups are alike in everything except the treatment. Surveys, scatter plots and past data can only show correlations.
5.
In the placebo group of a trial, $40$ of $250$ patients recovered without medicine. What percent of the placebo group is that? (Enter the number only.)
Answer: 16
$\frac{40}{250} \times 100 = 16$ , so $16\%$ recovered. This is why trials need a control group: some patients get better even without the medicine.
6.
Town A has many bike lanes, $50\,000$ residents and $100$ bike accidents per year. Town B has few bike lanes, $20\,000$ residents and $60$ bike accidents per year. A headline says: "Bike lanes cause accidents!" How many fewer bike accidents per $1000$ residents does Town A have?
Answer: 1
1. How many bike accidents per $1000$ residents does Town A have? 2
2. How many bike accidents per $1000$ residents does Town B have? 3
3. How many fewer accidents per $1000$ residents does Town A have? 1
7.
People who take painkillers more often report more headaches. Which explanation is most likely?
1. a)The correlation must be negative
2. b)Frequent headaches cause people to take more painkillers
3. c)Painkillers and headaches are not related at all
4. d)Painkillers cause headaches
Answer: Frequent headaches cause people to take more painkillers
This is reverse causation: people take painkillers because they have headaches. The headaches cause the painkillers, not the other way around.
8.
A researcher tests $300$ pairs of unrelated variables. Each pair has a $5\%$ chance of looking strongly correlated just by luck. How many strong-looking correlations should she expect?
Answer: 15
$300 \times 0.05 = 15$ . About $15$ correlations will look strong by pure coincidence, although none of the pairs is really connected.
9.
To test whether music helps studying, $24$ students are split at random into two equal groups. One group studies with music, the other in silence. On the test, the music group scores $936$ points in total and the silence group $864$ points in total. How many points higher is the average score of the music group?
Answer: 6
1. How many students are in each group? 12
2. What is the average score of the music group? 78
3. What is the average score of the silence group? 72
4. How many points higher is the music group's average? 6
10.
More firefighters are sent to bigger fires, and bigger fires cause more damage. So the number of firefighters at a fire is positively correlated with the damage. What is the confounding variable?
1. a)The day of the week
2. b)The size of the fire
3. c)The firefighters themselves
4. d)The color of the fire trucks
Answer: The size of the fire
The size of the fire drives both: a big fire needs more firefighters and also causes more damage. The firefighters do not cause the damage; they reduce it.
11.
In a randomized experiment, $12\%$ of the treatment group and $20\%$ of the control group got the flu. By how many percentage points is the rate lower in the treatment group?
Answer: 8
$20 - 12 = 8$ percentage points. Because the groups were formed at random, this difference estimates the effect of the treatment.
12.
An online shop shows version A of its website to $1500$ randomly chosen visitors and version B to another $1500$ . With version A, $45$ visitors buy something; with version B, $75$ do. By how many percentage points is the buying rate higher with version B?
Answer: 2
1. What percent of the version A visitors buy something? 3
2. What percent of the version B visitors buy something? 5
3. By how many percentage points is version B higher? 2
13.
$400$ volunteers are split at random into $4$ equal groups: one placebo group and three groups with different doses of a cold medicine. In the placebo group, $30$ people catch a cold; in the highest-dose group, $12$ people do. By how many percentage points is the cold rate lower in the highest-dose group?
Answer: 18
1. How many people are in each group? 100
2. What percent of the placebo group catch a cold? 30
3. What percent of the highest-dose group catch a cold? 12
4. By how many percentage points is the highest-dose rate lower? 18
14.
Which statement is correct?
1. a)A correlation of $r = 0.98$ proves that $x$ causes $y$
2. b)If two variables are not strongly correlated, one can never cause the other
3. c)A strong correlation alone never proves causation, and a cause can exist even when the correlation is weak
4. d)Correlation and causation mean the same thing when the sample is large
Answer: A strong correlation alone never proves causation, and a cause can exist even when the correlation is weak
A strong correlation can come from reverse causation, a confounding variable or coincidence, so it never proves causation alone. And a real cause can give a weak correlation when other variables hide it or the relationship is not a straight line. A large sample does not change this.
15.
$600$ volunteers are split at random into two equal groups. In the vaccine group, $27$ people get the flu; in the placebo group, $45$ people do. By how many percentage points is the flu rate lower in the vaccine group?
Answer: 6
Each group: $600 \div 2 = 300$ . Vaccine: $\frac{27}{300} \times 100 = 9\%$ . Placebo: $\frac{45}{300} \times 100 = 15\%$ . Difference: $15 - 9 = 6$ percentage points.
16.
Hospital A has an overall death rate of $3.4\%$ and Hospital B of $2.6\%$ , so B looks better. But Hospital A treats many more severely ill patients. Hospital A had $800$ severe patients, of whom $32$ died, and $200$ mild patients, of whom $2$ died. Hospital B had $200$ severe patients, of whom $10$ died, and $800$ mild patients, of whom $16$ died. For severe patients, by how many percentage points is Hospital A's death rate lower than Hospital B's?
Answer: 1
1. Check Hospital A's overall death rate: what is $\frac{32 + 2}{1000} \times 100$ ? 3.4
2. What is Hospital A's death rate for severe patients, in percent? 4
3. What is Hospital B's death rate for severe patients, in percent? 5
4. By how many percentage points is Hospital A lower for severe patients? 1
17.
A lifeguard records $100$ summer days. On the $50$ days with high ice cream sales there were $7$ swimmer rescues, and on the $50$ days with low sales there were $4$ . Split by weather: there were $30$ hot days with high sales ( $6$ rescues) and $10$ hot days with low sales ( $2$ rescues), $20$ cool days with high sales ( $1$ rescue) and $40$ cool days with low sales ( $2$ rescues). On hot days only, what is the difference in rescues per day between high-sales days and low-sales days?
Answer: 0
1. How many rescues per day were there on all high-sales days? 0.14
2. How many rescues per day were there on all low-sales days? 0.08
3. How many rescues per day were there on hot high-sales days? 0.2
4. How many rescues per day were there on hot low-sales days? 0.2
5. What is the difference on hot days? 0
18.
A researcher tests $20$ foods that have no real link to a disease. For each food there is a $5\%$ chance that the test shows a false "link" just by luck, and the tests are independent. What is the probability that at least one food shows a false link? Give it in percent, rounded to a whole number.
Answer: 64
1. What is the probability that one test shows no false link? Write it as a decimal. 0.95
2. What is the probability that all $20$ tests show no false link? Round to two decimal places. 0.36
3. What is the probability of at least one false link, as a decimal? 0.64
4. What is that in percent? 64