Correlation vs Causation

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Answer Key & Explanations

Correlation vs Causation · Grade 11 · Worksheet 1

1. Olivia is a data analyst for a national park service. She analyzes data from 75 forest regions and finds a strong positive correlation (r = 0.73) between the number of campfires reported each month and the number of search-and-rescue operations conducted that month. The park director proposes a new policy to ban all campfires, arguing that campfires directly cause hikers to get lost and require rescue. Explain why the director's causal conclusion is likely flawed and identify a plausible confounding variable that could explain this correlation. Answer: The director's conclusion is flawed because correlation does not imply causation. A likely confounding variable is the number of visitors per month: months with more visitors lead to more campfires and also more search-and-rescue operations, without campfires directly causing rescues. Solution: The data shows r = 0.73, which is a strong positive correlation between campfires and search-and-rescue operations. This means that as one variable increases, the other tends to increase as well.
   Full step-by-step solution

   Step 1: Understand the observed correlation. The data shows r = 0.73, which is a strong positive correlation between campfires and search-and-rescue operations. This means that as one variable increases, the other tends to increase as well. Step 2: Recognize that correlation does not equal causation. The director assumes that campfires cause more rescues, but correlation alone cannot prove causation. There may be a hidden variable (confounding variable) that influences both. Step 3: Identify a plausible confounding variable. The number of visitors to the park each month is a strong candidate. During peak tourist months, more people visit the park, leading to more campfires (since more people are camping) and also more search-and-rescue operations (since more people means more opportunities for accidents or getting lost). Step 4: Explain why this alternative explanation is more plausible. The relationship between campfires and rescues is likely indirect. Both are influenced by visitor volume. If the park director bans campfires, the number of rescues may not decrease if visitor numbers remain high, because hikers can still get lost without campfires. Step 5: Final conclusion. The director's causal conclusion is flawed because a confounding variable—monthly visitor numbers—likely drives both variables. Without controlling for this factor, one cannot claim that campfires cause more search-and-rescue operations.

2. A study of 27 coastal towns finds a Pearson correlation coefficient of r = 0.82 between the number of beach umbrellas sold (x) and the number of sunburn cases (y) in July. The mean of x is 62 with standard deviation 12, and the mean of y is 37 with standard deviation 7. Calculate the slope of the least-squares regression line for predicting sunburn cases from beach umbrella sales, then explain why this strong correlation does not imply that beach umbrellas cause sunburns. Answer: 0.47833 Solution: Identify given values: r = 0.82, sx = 12, sy = 7. Slope formula: b = r * (sy / sx) = 0.82 * (7/12). Calculate 7/12 = 0.58333...
   Full step-by-step solution

   Step 1: Identify given values: r = 0.82, sx = 12, sy = 7. Step 2: Slope formula: b = r * (sy / sx) = 0.82 * (7/12). Step 3: Calculate 7/12 = 0.58333... Step 4: Multiply: 0.82 * 0.58333 = 0.47833 (rounded to 5 decimal places). Step 5: The slope means that for each additional beach umbrella sold, sunburn cases are predicted to increase by about 0.478. Step 6: Correlation does not equal causation. A lurking variable, such as hot sunny weather, causes more people to buy beach umbrellas (to seek shade) and also causes more people to be exposed to the sun, leading to more sunburns. The beach umbrellas themselves do not cause sunburns; both variables are associated with the same external factor (intense sunlight). The answer is slope = 0.47833.

3. Mere collects data from 15 schools on the number of hours of weekly music practice (x) and the average GPA (y) of students. The Pearson correlation coefficient is r = 0.82. The mean of x is 6.5 hours with standard deviation 2.4 hours, and the mean of y is 3.2 with standard deviation 0.6. Calculate the slope of the least-squares regression line for predicting GPA from music practice hours. Then, explain why this strong correlation does not imply that more music practice causes higher GPA. Answer: Slope = 0.205 Solution: Identify given values: r = 0.82, sx = 2.4, sy = 0.6. Slope formula: b = r * (sy / sx) = 0.82 * (0.6 / 2.4). Simplify the fraction: 0.6 / 2.4 = 0.25.
   Full step-by-step solution

   Step 1: Identify given values: r = 0.82, sx = 2.4, sy = 0.6. Step 2: Slope formula: b = r * (sy / sx) = 0.82 * (0.6 / 2.4). Step 3: Simplify the fraction: 0.6 / 2.4 = 0.25. Step 4: Multiply: 0.82 * 0.25 = 0.205. Step 5: The slope means that for each additional hour of weekly music practice, GPA is predicted to increase by 0.205. Step 6: A strong correlation (r = 0.82) indicates a linear relationship, but it does not prove causation. A lurking variable such as family income or school resources could cause both more music practice (due to access to lessons) and higher GPA (due to better educational support). Without a controlled experiment, we cannot conclude that music practice causes higher GPA. The answer is slope = 0.205.

4. Matiu collects data on the number of hours of television watched per week (x) and the number of books read per year (y) for 10 adults. The Pearson correlation coefficient is r = -0.82. The mean of x is 18 hours with standard deviation 4, and the mean of y is 12 books with standard deviation 3. Calculate the slope of the least-squares regression line for predicting books read from television hours, then explain why this negative correlation does not imply that watching television causes people to read fewer books. Answer: Slope = -0.615 Solution: Identify given values: r = -0.82, sx = 4, sy = 3. Slope formula: b = r * (sy / sx) = -0.82 * (3/4) = -0.82 * 0.75 = -0.615.
   Full step-by-step solution

   Step 1: Identify given values: r = -0.82, sx = 4, sy = 3. Step 2: Slope formula: b = r * (sy / sx) = -0.82 * (3/4) = -0.82 * 0.75 = -0.615. Step 3: The slope means that for each additional hour of television watched per week, the predicted number of books read per year decreases by about 0.615. Step 4: Correlation does not equal causation. A lurking variable, such as a person's age or education level, could cause both higher television watching and lower book reading. For example, older adults might watch more television and read fewer books due to declining eyesight, or people with less education might prefer television over reading. The negative correlation shows an association, but without a controlled experiment, we cannot conclude that watching television causes people to read fewer books. The answer is slope = -0.615.

5. Sophia, a data scientist at a tech company, analyzes employee data and finds a strong positive correlation (r = 0.91) between the number of years of experience an employee has and their annual salary. The company's management concludes that increasing an employee's years of experience directly causes a higher salary, so they propose a policy that all new hires must have at least 10 years of experience to maximize salary potential. Explain why this causal conclusion is flawed, and identify a likely confounding variable that could account for the observed correlation. Answer: The correlation does not imply causation; a confounding variable such as job performance, education level, or industry demand could influence both experience and salary. Solution: Recognize the correlation coefficient r = 0.91 indicates a strong positive linear relationship between years of experience and salary. This means as experience increases, salary tends to increase.
   Full step-by-step solution

   Step 1: Recognize the correlation coefficient r = 0.91 indicates a strong positive linear relationship between years of experience and salary. This means as experience increases, salary tends to increase. Step 2: Understand that correlation does not prove causation. A strong correlation can occur due to: (a) a direct causal relationship (which the management assumes), (b) a reverse causal relationship (higher salary allows employees to stay longer and gain experience), (c) a confounding variable that influences both variables. Step 3: Identify a plausible confounding variable. For example, job performance: employees with higher performance ratings may receive promotions (increasing experience) and also get salary raises. Alternatively, education level: employees with advanced degrees may have both longer careers (experience) and higher starting salaries. Another is industry demand: certain high-demand fields offer both rapid experience accumulation and high salaries. Step 4: Conclude that the management's policy is flawed because it ignores these alternative explanations. Simply requiring 10 years of experience does not guarantee a high salary, as other factors are at play. The final answer is: The correlation does not imply causation; a confounding variable such as job performance, education level, or industry demand could influence both experience and salary.