Correlation vs Causation

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Correlation vs Causation · Grade 11 · Worksheet 3

1. Noah collects data on the average daily temperature (in degrees Celsius) and the number of ice cream cones sold at a beachfront shop for 16 consecutive days in July. A scatter plot of the data shows a strong positive linear relationship. The correlation coefficient is r = 0.91. The regression line equation is ŷ = 11x + 21, where x represents the average daily temperature in degrees Celsius and ŷ represents the predicted number of ice cream cones sold. On a day when the average temperature was 26 degrees Celsius, the shop actually sold 306 ice cream cones. Based on this high correlation and the data, can we conclude that increasing the average daily temperature causes an increase in ice cream cone sales? Explain why or why not, referencing a potential confounding variable in this context. Answer: No, we cannot conclude causation because correlation does not imply causation. A potential confounding variable is the season (summer), which causes both higher temperatures and more people going to the beach, leading to increased ice cream sales. Solution: Identify the given information. We have a strong positive correlation (r = 0.91) between average daily temperature (x) and ice cream cone sales (y). The regression line is ŷ = 11x + 21.
   Full step-by-step solution

   Step 1: Identify the given information. We have a strong positive correlation (r = 0.91) between average daily temperature (x) and ice cream cone sales (y). The regression line is ŷ = 11x + 21. Step 2: Understand the question. The problem asks if we can conclude that increasing temperature causes an increase in ice cream cone sales based on this correlation. We must explain why or why not. Step 3: Recognize the key statistical principle. A high correlation coefficient (r = 0.91) indicates a strong linear relationship between two variables. However, correlation does NOT imply causation. Just because two variables are strongly correlated does not mean one causes the other to change. Step 4: Identify potential confounding variables. A confounding variable is a third variable that influences both the independent variable (temperature) and the dependent variable (ice cream sales). In this context, the season (summer) is a likely confounding variable. During summer: - Temperatures are naturally higher. - More people visit the beach, increasing the number of potential customers. - People are more likely to want cold treats like ice cream. Step 5: Explain why causation cannot be concluded. The increase in ice cream sales is likely due to the combination of higher temperatures AND more people being at the beach (a result of summer), not just the temperature alone. Without controlling for this confounding variable (e.g., through a controlled experiment), we cannot claim that temperature causes higher sales. Step 6: Conclusion. No, we cannot conclude that increasing temperature causes an increase in ice cream cone sales. The strong correlation (r = 0.91) is likely driven by the confounding variable of the summer season, which affects both temperature and beach attendance, and therefore ice cream sales. A controlled experiment or further analysis (e.g., comparing sales on hot vs. cold days with the same number of beach visitors) would be needed to establish causation.

2. Aroha, a data analyst for a national park service, has collected data from 35 different hiking trails. She finds a strong positive correlation (r = 0.79) between the number of trail markers installed on a trail and the number of visitor injuries reported on that trail over the past year. Based on this finding, a park manager suggests removing trail markers to reduce injuries. Explain why this causal conclusion is flawed, and identify a likely confounding variable that could explain the observed correlation. Answer: Correlation does not imply causation. A likely confounding variable is trail difficulty or popularity: more difficult or popular trails have more markers (for safety) and also see more injuries due to the challenging terrain or higher visitor numbers. Solution: Recognize that a correlation of r = 0.79 indicates a strong positive linear relationship between two variables: trail markers and injuries. As one increases, the other tends to increase.
   Full step-by-step solution

   Step 1: Recognize that a correlation of r = 0.79 indicates a strong positive linear relationship between two variables: trail markers and injuries. As one increases, the other tends to increase. Step 2: Understand that correlation measures association, not causation. Even a strong correlation does not prove that changing one variable will change the other. Step 3: Identify potential confounding variables. In this scenario, a likely confounder is trail difficulty or popularity. More difficult trails (e.g., steep, rocky, long) often require more trail markers for navigation and safety. These same difficult trails also pose greater physical risk to hikers, leading to more injuries. Similarly, very popular trails may have more markers due to higher maintenance budgets and also see more injuries simply because more people use them. Step 4: Conclude that the manager's suggestion to remove markers is based on a flawed causal assumption. The correlation is likely driven by a third variable (trail difficulty or popularity) rather than a direct causal link from markers to injuries. Answer: Correlation does not imply causation. A likely confounding variable is trail difficulty or popularity: more difficult or popular trails have more markers (for safety) and also see more injuries due to the challenging terrain or higher visitor numbers.

3. Sophia, a data analyst for a large school district, examines the relationship between the number of books in a school's library and the average standardized test scores of its students. She collects data from 50 high schools and calculates a correlation coefficient of r = 0.78. The school board, upon seeing this strong positive correlation, immediately proposes a policy to double the number of books in every school library, claiming it will directly cause a significant increase in test scores. As a critical thinker, explain why the school board's causal conclusion is flawed. Identify a likely confounding variable that could explain the observed correlation, and describe what additional evidence or study design would be needed to establish a true causal relationship between library books and test scores. Answer: The school board's conclusion is flawed because correlation does not imply causation. A likely confounding variable is socioeconomic status (SES). Schools in wealthier districts can afford more books and also tend to have higher test scores due to better resources, smaller class sizes, and more parental support. To establish causation, a controlled experiment is needed where schools are randomly assigned to receive more books or not, while controlling for other factors like SES, teacher quality, and funding. Solution: A correlation of r = 0.78 indicates a strong positive linear relationship between the number of library books and test scores. As one variable increases, the other tends to increase as well.
   Full step-by-step solution

   Step 1: Understand the correlation. A correlation of r = 0.78 indicates a strong positive linear relationship between the number of library books and test scores. As one variable increases, the other tends to increase as well. However, this only describes an association, not a cause-and-effect mechanism. Step 2: Identify the flaw in the causal claim. The school board assumes that increasing library books will directly raise test scores. This is a classic example of the 'correlation does not imply causation' fallacy. The correlation could be due to: - A confounding variable: A third factor that influences both variables. - Reverse causation: Higher test scores might lead to more funding for libraries, not the other way around. - A spurious correlation: The relationship might be coincidental. Step 3: Identify a likely confounding variable. The most plausible confounder is socioeconomic status (SES) of the school community. Schools in high-SES areas typically have: - Larger budgets for library resources. - Students with more educational support at home. - Better overall school facilities and teacher quality. These factors, not just the number of books, drive higher test scores. Step 4: Describe the additional evidence needed. To prove causation, the school board would need to: 1. Conduct a randomized controlled trial (RCT): Randomly assign a group of schools to receive a significant increase in library books (treatment group) and another group to receive no change (control group). Random assignment helps balance confounding variables like SES across groups. 2. Measure test scores before and after the intervention to see if the treatment group improves more than the control group. 3. Control for other variables statistically (e.g., using multiple regression) if randomization is not possible, by measuring and adjusting for SES, teacher-student ratio, and prior test scores. 4. Establish temporal precedence: Ensure the increase in books happens before any observed change in test scores. Step 5: Conclusion. The correlation (r = 0.78) alone is insufficient to claim causation. The school board's policy is based on a flawed interpretation of data. The most likely explanation is that a confounding variable, such as socioeconomic status, drives both variables. A controlled experiment or rigorous statistical control is required to establish a causal link.

4. Emma collects data on the number of hours of sunlight per day (x) and the number of ice cream cones sold (y) at her local shop over 7 days. The data yields a Pearson correlation coefficient of r = 0.95. Does this strong correlation prove that more sunlight causes more ice cream sales? Explain why or why not, and identify a possible confounding variable. Answer: No, correlation does not imply causation. A possible confounding variable is higher temperature, which increases both sunlight hours and ice cream demand. Solution: The correlation coefficient r = 0.95 indicates a very strong positive linear relationship between sunlight hours and ice cream sales. Step 2: However, correlation alone does not prove causation.
   Full step-by-step solution

   Step 1: The correlation coefficient r = 0.95 indicates a very strong positive linear relationship between sunlight hours and ice cream sales. Step 2: However, correlation alone does not prove causation. There could be a lurking or confounding variable. Step 3: In this context, temperature is a likely confounding variable: longer sunlight hours often coincide with warmer weather, and warmer weather increases people's desire for ice cream. Step 4: Therefore, the observed correlation may be due to the common cause of temperature, not a direct causal link from sunlight to sales. Step 5: To establish causation, a controlled experiment (e.g., randomly assigning sunlight exposure) or more advanced statistical methods (e.g., regression with temperature as a control variable) would be needed. Final answer: No, correlation does not imply causation; temperature is a plausible confounding variable.

5. ∫(4x³ - 6x² + 2)dx from 1 to 3 = ? Answer: 32 Solution: Find the antiderivative of 4x³ - 6x² + 2 Antiderivative = (4/4)x⁴ - (6/3)x³ + 2x = x⁴ - 2x³ + 2x Evaluate at the upper limit (x = 3) F(3) = (3)⁴ - 2(3)³ + 2(3) = 81 - 54 + 6 = 33 Evaluate at the lower limit (x = 1) F(1) = (1)⁴ - 2(1)³ + 2(1) = 1 - 2 + 2 = 1 Subtract: F(3) - F(1) = 33 - 1 = 32…
   Full step-by-step solution

   Step 1: Find the antiderivative of 4x³ - 6x² + 2 Antiderivative = (4/4)x⁴ - (6/3)x³ + 2x = x⁴ - 2x³ + 2x Step 2: Evaluate at the upper limit (x = 3) F(3) = (3)⁴ - 2(3)³ + 2(3) = 81 - 54 + 6 = 33 Step 3: Evaluate at the lower limit (x = 1) F(1) = (1)⁴ - 2(1)³ + 2(1) = 1 - 2 + 2 = 1 Step 4: Subtract: F(3) - F(1) = 33 - 1 = 32 The answer is 32.