Correlation Coefficient

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Answer Key & Explanations

Correlation Coefficient · Grade 11 · Worksheet 1

1. Emma, a botanist, is studying the relationship between the number of hours of sunlight per day and the height (in cm) of a particular species of plant after 30 days. She collects data from 9 plants and calculates the Pearson correlation coefficient, finding r = -0.73. Interpret this correlation coefficient in the context of the study, describing both the strength and direction of the linear relationship between sunlight hours and plant height. Answer: There is a strong negative linear relationship between the number of hours of sunlight per day and plant height; as sunlight hours increase, plant height tends to decrease strongly. Solution: Interpret the direction. The correlation coefficient r = -0.73 is negative, meaning that as the number of hours of sunlight per day increases, the plant height after 30 days tends to decrease.
   Full step-by-step solution

   Step 1: Interpret the direction. The correlation coefficient r = -0.73 is negative, meaning that as the number of hours of sunlight per day increases, the plant height after 30 days tends to decrease. This indicates a negative linear relationship. Step 2: Interpret the strength. The absolute value of r is | -0.73 | = 0.73. Using standard guidelines for correlation strength: - 0.00 to 0.30: weak correlation - 0.30 to 0.70: moderate correlation - 0.70 to 1.00: strong correlation Since 0.73 is greater than 0.70, this indicates a strong strength relationship. Step 3: Combine direction and strength. The relationship is a strong negative linear relationship. In context: plants receiving more sunlight per day tend to have a strongly lower height after 30 days, but the relationship is not perfect, meaning other factors (like soil quality or water) also influence plant height. Final answer: There is a strong negative linear relationship between the number of hours of sunlight per day and plant height; as sunlight hours increase, plant height tends to decrease strongly.

2. Mere, a kaitiaki (environmental guardian), is studying the relationship between the age of a native forest (in years since regeneration began) and the number of endemic bird species observed in that forest. She collects data from 10 forest plots and calculates the following summary statistics: the mean age of the forests is 45 years, the mean number of bird species is 18, the standard deviation of age is 12 years, the standard deviation of bird species is 5, and the sum of the products of z-scores (Σzₓzᵧ) is 6.3. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between forest age and endemic bird species count. Answer: r = 0.70, indicating a strong positive linear relationship between forest age and the number of endemic bird species. Solution: Recall the formula for the Pearson correlation coefficient using the sum of products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 6.3 and n = 10. Substitute the values into the formula: r = 6.3 / (10 - 1) = 6.3 / 9.
   Full step-by-step solution

   Step 1: Recall the formula for the Pearson correlation coefficient using the sum of products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 6.3 and n = 10. Step 2: Substitute the values into the formula: r = 6.3 / (10 - 1) = 6.3 / 9. Step 3: Calculate the quotient: 6.3 ÷ 9 = 0.7. Step 4: Interpret the direction: Since r = 0.7 is positive, there is a positive linear relationship. This means that as forest age increases, the number of endemic bird species also tends to increase. Step 5: Interpret the strength: The absolute value |r| = 0.7. Using standard guidelines (|r| < 0.3 is weak, 0.3 ≤ |r| < 0.7 is moderate, |r| ≥ 0.7 is strong), this indicates a strong linear relationship. Final Answer: r = 0.70, indicating a strong positive linear relationship between forest age and the number of endemic bird species.

3. Noah, a soil scientist, is studying the relationship between the organic matter content (as a percentage of dry soil mass) and the water holding capacity (in milliliters of water per 100 grams of soil) for 11 different soil samples. He calculates the following summary statistics: the mean organic matter content is 8.5%, the mean water holding capacity is 42 mL/100g, the standard deviation of organic matter is 2.2%, the standard deviation of water holding capacity is 9.6 mL/100g, and the sum of the products of the z-scores (Σzₓzᵧ) is 8.8. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between organic matter content and water holding capacity. Answer: r = 0.88, indicating a strong positive linear relationship between organic matter content and water holding capacity. Solution: Recall the formula for the Pearson correlation coefficient using the sum of the products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 8.8 and n = 11.
   Full step-by-step solution

   Step 1: Recall the formula for the Pearson correlation coefficient using the sum of the products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 8.8 and n = 11. Step 2: Substitute the values into the formula: r = 8.8 / (11 - 1) = 8.8 / 10. Step 3: Calculate the quotient: 8.8 ÷ 10 = 0.88. Step 4: Interpret the direction: Since r = 0.88 is positive, there is a positive linear relationship. This means that as the organic matter content in the soil increases, the water holding capacity also tends to increase. Step 5: Interpret the strength: The absolute value |r| = 0.88 is greater than 0.7. Using standard guidelines (|r| < 0.3 is weak, 0.3 ≤ |r| < 0.7 is moderate, |r| ≥ 0.7 is strong), this indicates a strong linear relationship. Final Answer: r = 0.88, indicating a strong positive linear relationship between organic matter content and water holding capacity.

4. A research team is studying the relationship between study hours and exam scores for Grade 11 students. They collected data from 30 students and calculated a correlation coefficient of r = 0.72. The team wants to determine if this correlation is statistically significant at the α = 0.05 level. If the critical value for a two-tailed test with 28 degrees of freedom is approximately 0.361, what conclusion should the researchers draw about the relationship between study hours and exam scores? Answer: The correlation is statistically significant because the calculated correlation coefficient (0.72) is greater than the critical value (0.361). Solution: - Sample size n = 30 - Correlation coefficient r = 0.72 - Significance level α = 0.05 - Degrees of freedom df = n - 2 = 28 - Critical value from table = 0.361 We need to check if the correlation is statistically significant.
   Full step-by-step solution

   Step 1: Understand the problem We are given: - Sample size n = 30 - Correlation coefficient r = 0.72 - Significance level α = 0.05 - Two-tailed test - Degrees of freedom df = n - 2 = 28 - Critical value from table = 0.361 We need to check if the correlation is statistically significant. --- Step 2: Recall the decision rule for correlation significance For a two-tailed test, we compare the absolute value of the calculated r with the critical value from the table. If |r| > critical value, then the correlation is statistically significant at the given α level. --- Step 3: Apply the numbers Calculated r = 0.72 Critical value = 0.361 Since |0.72| = 0.72 and 0.72 > 0.361, the calculated correlation coefficient exceeds the critical value. --- Step 4: Draw the conclusion Because the calculated r is greater than the critical value, we reject the null hypothesis (which states that there is no correlation in the population). Thus, the correlation between study hours and exam scores is statistically significant at α = 0.05. --- Final Answer: The correlation is statistically significant because the calculated correlation coefficient (0.72) is greater than the critical value (0.361).

5. Given ∑x = 15, ∑y = 30, ∑xy = 110, ∑x² = 55, ∑y² = 220, n = 5, compute r = ? Answer: 1 Solution: Write the Pearson correlation formula: r = [n∑xy - (∑x)(∑y)] / sqrt([n∑x² - (∑x)²][n∑y² - (∑y)²]) Calculate the numerator: n∑xy - (∑x)(∑y) = (5)(110) - (15)(30) = 550 - 450 = 100 Calculate the x-part of denominator: n∑x² - (∑x)² = (5)(55) - (15)² = 275 - 225 = 50 Calculate the y-part of…
   Full step-by-step solution

   Step 1: Write the Pearson correlation formula: r = [n∑xy - (∑x)(∑y)] / sqrt([n∑x² - (∑x)²][n∑y² - (∑y)²]) Step 2: Calculate the numerator: n∑xy - (∑x)(∑y) = (5)(110) - (15)(30) = 550 - 450 = 100 Step 3: Calculate the x-part of denominator: n∑x² - (∑x)² = (5)(55) - (15)² = 275 - 225 = 50 Step 4: Calculate the y-part of denominator: n∑y² - (∑y)² = (5)(220) - (30)² = 1100 - 900 = 200 Step 5: Multiply the denominator parts: 50 × 200 = 10000 Step 6: Take square root of denominator: sqrt(10000) = 100 Step 7: Divide numerator by denominator: 100 ÷ 100 = 1 The answer is 1.