Correlation Coefficient

lessonbunny.com

Answer Key & Explanations

Correlation Coefficient · Grade 11 · Worksheet 2

1. Mason, a hydrologist, is studying the relationship between the depth of a river (in meters) at a specific point and the water flow velocity (in meters per second) at that same point. He collects data from 12 different river locations and computes the following summary statistics: the mean depth is 2.7 meters, the mean velocity is 1.2 meters per second, the standard deviation of depth is 0.7 meters, the standard deviation of velocity is 0.4 meters per second, and the sum of the products of the z-scores (Σzₓzᵧ) is 7.7. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between river depth and water flow velocity. Answer: r = 0.70, indicating a strong positive linear relationship between river depth and water flow velocity. Solution: Recall the formula for the Pearson correlation coefficient using the sum of products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 7.7 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 = 7.7 and n = 12. Step 2: Substitute the values into the formula: r = 7.7 / (12 - 1) = 7.7 / 11. Step 3: Calculate the quotient: 7.7 ÷ 11 = 0.7 exactly (since 7.7/11 = 77/110 = 7/10 = 0.7). Step 4: Interpret the direction: Since r = 0.7 is positive, there is a positive linear relationship. This means that as river depth increases, water flow velocity 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 river depth and water flow velocity.

2. Tane, a sports scientist, is studying the relationship between the number of hours of sleep an athlete gets per night (x) and their reaction time in milliseconds (y) for 12 athletes. He collects the data and calculates the Pearson correlation coefficient, finding r = -0.71. Interpret this correlation coefficient in the context of the study, describing both the strength and direction of the linear relationship between sleep hours and reaction time. Answer: There is a strong negative linear relationship between hours of sleep and reaction time; as sleep hours increase, reaction time tends to decrease strongly. Solution: Interpret the direction. The correlation coefficient r = -0.71 is negative, meaning that as the number of hours of sleep per night increases, the reaction time tends to decrease.
   Full step-by-step solution

   Step 1: Interpret the direction. The correlation coefficient r = -0.71 is negative, meaning that as the number of hours of sleep per night increases, the reaction time tends to decrease. This indicates a negative linear relationship. Step 2: Interpret the strength. The absolute value of r is | -0.71 | = 0.71. 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 0.71 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 get more hours of sleep tend to have substantially faster reaction times (lower reaction time in milliseconds), and the relationship is strong, meaning sleep hours are a good predictor of reaction time, though other factors may also play a role. Final answer: There is a strong negative linear relationship between hours of sleep and reaction time; as sleep hours increase, reaction time tends to decrease strongly.

3. Matiu, a marine biologist, is investigating the relationship between water temperature (°C) and the number of fish observed per hour at a reef. He records data at 10 different times and calculates a Pearson correlation coefficient of r = 0.92. Interpret this correlation coefficient in the context of the study, including both the direction and strength of the linear relationship. Answer: There is a strong positive linear relationship between water temperature and the number of fish observed per hour. Solution: Interpret the direction of the relationship. The correlation coefficient is positive (r = 0.92). A positive correlation means that as one variable increases, the other tends to increase as well.
   Full step-by-step solution

   Step 1: Interpret the direction of the relationship. The correlation coefficient is positive (r = 0.92). A positive correlation means that as one variable increases, the other tends to increase as well. In this context, as water temperature increases, the number of fish observed per hour also tends to increase. Step 2: Interpret the strength of the relationship. The absolute value of the correlation coefficient is |r| = 0.92. General guidelines for strength: - 0.00 to 0.30: Weak linear relationship - 0.30 to 0.70: Moderate linear relationship - 0.70 to 1.00: Strong linear relationship Since 0.92 is greater than 0.70, this indicates a strong linear relationship. Step 3: Combine direction and strength. A strong positive linear relationship means that the data points are closely clustered around an upward-sloping line. Matiu can conclude that there is a strong positive linear relationship between water temperature and the number of fish observed per hour at the reef. Final Answer: There is a strong positive linear relationship between water temperature and the number of fish observed per hour.

4. A scatter plot shows the relationship between hours studied (x) and exam scores (y) for 20 students. The data points form an approximately linear pattern with a correlation coefficient of r = 0.85. If a student studies for 3 hours, what exam score would you predict using the least squares regression line? The regression equation is ŷ = 4.2x + 65. Answer: 77.6 Solution: Identify the regression equation. The problem gives the least squares regression line as: ŷ = 4.2x + 65 Understand what the variables represent. Here, ŷ is the predicted exam score, and x is the number of hours studied.
   Full step-by-step solution

   Step 1: Identify the regression equation. The problem gives the least squares regression line as: ŷ = 4.2x + 65 Step 2: Understand what the variables represent. Here, ŷ is the predicted exam score, and x is the number of hours studied. Step 3: Substitute the given hours studied into the equation. We are told the student studies for 3 hours, so x = 3. ŷ = 4.2 * (3) + 65 Step 4: Perform the multiplication. 4.2 * 3 = 12.6 Step 5: Add the result to the y-intercept. ŷ = 12.6 + 65 Step 6: Complete the addition. 12.6 + 65 = 77.6 Step 7: State the final prediction. The predicted exam score for a student who studies 3 hours is 77.6. The correlation coefficient (r = 0.85) is not needed for this calculation because the regression equation is already provided.

5. Mere, a kaitiaki (environmental guardian), is studying the relationship between the number of native trees planted per hectare (x) and the annual rainfall infiltration rate in millimeters per hour (y) across 11 different restoration sites. She calculates the following summary statistics: the mean number of trees planted per hectare is 240, the mean infiltration rate is 18 mm/hr, the standard deviation of tree count is 45 trees, the standard deviation of infiltration rate is 4 mm/hr, and the sum of the products of the z-scores (Σzₓzᵧ) is -7.2. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between the number of native trees planted per hectare and the annual rainfall infiltration rate. Answer: r = -0.72, indicating a strong negative linear relationship between the number of native trees planted per hectare and the annual rainfall infiltration rate. Solution: Recall the formula for the Pearson correlation coefficient using the sum of the products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = -7.2 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 = -7.2 and n = 11. Step 2: Substitute the values into the formula: r = (-7.2) / (11 - 1) = -7.2 / 10. Step 3: Calculate the quotient: -7.2 ÷ 10 = -0.72. Step 4: Interpret the direction: Since r = -0.72 is negative, there is a negative linear relationship. This means that as the number of native trees planted per hectare increases, the annual rainfall infiltration rate tends to decrease. Step 5: Interpret the strength: The absolute value |r| = 0.72 is greater than or equal to 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 = -0.72, indicating a strong negative linear relationship between the number of native trees planted per hectare and the annual rainfall infiltration rate.