Correlation vs Causation

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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
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.)
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?
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
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.)
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?
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
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?
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?
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
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?
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?
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?
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
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?
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?
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?
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.