Correlation Coefficient

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Answer Key & Explanations

Correlation Coefficient · Grade 11 · Worksheet 3

1. Emma, a sports scientist, is studying the relationship between the number of hours of training per week and the time (in seconds) to complete a 100-meter sprint for 12 athletes. She calculates the Pearson correlation coefficient and finds r = -0.85. Interpret this correlation coefficient in the context of the study, describing both the strength and direction of the linear relationship between training hours and sprint time. Answer: There is a strong negative linear relationship between training hours and sprint time; as training hours increase, sprint time tends to decrease strongly. Solution: Interpret the direction. The correlation coefficient r = -0.85 is negative, meaning that as the number of training hours per week increases, the sprint time tends to decrease.
   Full step-by-step solution

   Step 1: Interpret the direction. The correlation coefficient r = -0.85 is negative, meaning that as the number of training hours per week increases, the sprint time tends to decrease. This indicates a negative linear relationship. Step 2: Interpret the strength. The absolute value of r is | -0.85 | = 0.85. 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 0.85 falls within the strong range (0.70 to 1.00), this indicates a strong strength relationship. Step 3: Combine direction and strength. The relationship is a strong negative linear relationship. In context: athletes who train more hours per week tend to have substantially lower sprint times, but the relationship is not perfect, meaning other factors (such as genetics, diet, or technique) also influence performance. Final answer: There is a strong negative linear relationship between training hours and sprint time; as training hours increase, sprint time tends to decrease strongly.

2. Charlotte, an environmental scientist, is studying the relationship between the average daily temperature (in degrees Celsius) and the number of mosquitoes trapped per night in a wetland area. She collects data from 7 different nights and records the following summary statistics: the mean temperature is 22 degrees Celsius, the mean number of mosquitoes is 137, the standard deviation of temperature is 7 degrees Celsius, the standard deviation of mosquitoes is 12, and the sum of the products of the z-scores (Σzₓzᵧ) is 5.2. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between average daily temperature and mosquito count. Answer: r = 0.87, indicating a strong positive linear relationship between average daily temperature and mosquito count. Solution: Recall the formula for the Pearson correlation coefficient using the sum of the products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 5.2 and n = 7.
   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 = 5.2 and n = 7. Step 2: Substitute the values into the formula: r = 5.2 / (7 - 1) = 5.2 / 6. Step 3: Calculate the quotient: 5.2 ÷ 6 = 0.8666... which rounds to 0.87. Step 4: Interpret the direction: Since r = 0.87 is positive, there is a positive linear relationship. This means that as the average daily temperature increases, the number of mosquitoes trapped per night also tends to increase. Step 5: Interpret the strength: The absolute value |r| = 0.87 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 = 0.87, indicating a strong positive linear relationship between average daily temperature and mosquito count.

3. Mere, a marine ecologist, is studying the relationship between the surface water temperature (in degrees Celsius) and the concentration of dissolved oxygen (in mg/L) in a coastal estuary. She collects data from 13 different sampling stations and calculates the following summary statistics: the mean water temperature is 18.5°C, the mean dissolved oxygen concentration is 7.2 mg/L, the standard deviation of temperature is 3.2°C, the standard deviation of dissolved oxygen is 1.5 mg/L, and the sum of the products of the z-scores (Σzₓzᵧ) is 11.2. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between water temperature and dissolved oxygen concentration. Answer: r = 0.93, indicating a strong positive linear relationship between water temperature and dissolved oxygen concentration. Solution: Recall the formula for the Pearson correlation coefficient using the sum of products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 11.2 and n = 13.
   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 = 11.2 and n = 13. Step 2: Substitute the values into the formula: r = 11.2 / (13 - 1) = 11.2 / 12. Step 3: Calculate the quotient: 11.2 ÷ 12 = 0.93333..., which rounds to 0.93. Step 4: Interpret the direction: Since r = 0.93 is positive, there is a positive linear relationship. This means that as water temperature increases, dissolved oxygen concentration also tends to increase. Step 5: Interpret the strength: The absolute value |r| = 0.93 is much 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 = 0.93, indicating a strong positive linear relationship between water temperature and dissolved oxygen concentration.

4. Mason, a climatologist, is studying the relationship between the average number of frost-free days per year (x) and the average annual apple yield in tonnes per hectare (y) for 12 orchard regions. He calculates the following summary statistics: the mean of x is 172 days, the mean of y is 27 tonnes/hectare, the standard deviation of x is 22 days, the standard deviation of y is 7 tonnes/hectare, and the sum of the products of z-scores (Σzₓzᵧ) is 8.8. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between frost-free days and apple yield. Answer: r = 0.80, indicating a strong positive linear relationship between frost-free days and apple yield. Solution: Recall the formula for the Pearson correlation coefficient using the sum of products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 8.8 and n = 12.
   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 = 8.8 and n = 12. Step 2: Substitute the values into the formula: r = 8.8 / (12 - 1) = 8.8 / 11. Step 3: Calculate the quotient: 8.8 ÷ 11 = 0.8 exactly. Step 4: Interpret the direction: Since r = 0.8 is positive, there is a positive linear relationship; as the average number of frost-free days increases, the average annual apple yield tends to increase. Step 5: Interpret the strength: The absolute value |r| = 0.8 is greater than 0.7, so the relationship is strong. Final Answer: r = 0.80, indicating a strong positive linear relationship between frost-free days per year and apple yield.

5. Given the data points (1, 2), (2, 4), (3, 6), (4, 8), compute the Pearson correlation coefficient r = ? Answer: 1 Solution: We have data points: (1, 2), (2, 4), (3, 6), (4, 8).
   Full step-by-step solution

   Let's compute the Pearson correlation coefficient r step by step. We have data points: (1, 2), (2, 4), (3, 6), (4, 8). --- **Step 1: List the x and y values** x: 1, 2, 3, 4 y: 2, 4, 6, 8 n = 4 (number of pairs) --- **Step 2: Compute the means** mean_x = (1 + 2 + 3 + 4) / 4 = 10 / 4 = 2.5 mean_y = (2 + 4 + 6 + 8) / 4 = 20 / 4 = 5 --- **Step 3: Compute deviations and products** We will use the formula: r = [ sum( (x_i - mean_x) * (y_i - mean_y) ) ] / [ sqrt( sum( (x_i - mean_x)^2 ) * sum( (y_i - mean_y)^2 ) ) ] First, compute (x_i - mean_x) and (y_i - mean_y): For (1, 2): dx = 1 - 2.5 = -1.5 dy = 2 - 5 = -3 product = (-1.5)*(-3) = 4.5 dx^2 = 2.25 dy^2 = 9 For (2, 4): dx = 2 - 2.5 = -0.5 dy = 4 - 5 = -1 product = (-0.5)*(-1) = 0.5 dx^2 = 0.25 dy^2 = 1 For (3, 6): dx = 3 - 2.5 = 0.5 dy = 6 - 5 = 1 product = 0.5 * 1 = 0.5 dx^2 = 0.25 dy^2 = 1 For (4, 8): dx = 4 - 2.5 = 1.5 dy = 8 - 5 = 3 product = 1.5 * 3 = 4.5 dx^2 = 2.25 dy^2 = 9 --- **Step 4: Sum the necessary values** sum of products = 4.5 + 0.5 + 0.5 + 4.5 = 10 sum of dx^2 = 2.25 + 0.25 + 0.25 + 2.25 = 5 sum of dy^2 = 9 + 1 + 1 + 9 = 20 --- **Step 5: Compute r** r = 10 / sqrt(5 * 20) = 10 / sqrt(100) = 10 / 10 = 1 --- **Step 6: Interpretation** The Pearson correlation coefficient r = 1 means there is a perfect positive linear relationship between x and y. Indeed, y = 2x exactly for all points. --- **Final Answer:** 1