Correlation Coefficient

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Answer Key & Explanations

Correlation Coefficient · Grade 11 · Worksheet 3

1. Matiu recorded the number of hours he spent on creative writing per week (x) and the number of pages he wrote (y) over 6 weeks. The data are shown in the scatter plot: (12, 18), (14, 22), (16, 26), (18, 30), (20, 34), (22, 38). Calculate the Pearson correlation coefficient r for this data, and interpret its strength and direction. Answer: 1.000 Solution: Compute the sums. n = 6.
   Full step-by-step solution

   Step 1: Compute the sums. n = 6. Σx = 12 + 14 + 16 + 18 + 20 + 22 = 102 Σy = 18 + 22 + 26 + 30 + 34 + 38 = 168 Σxy = 12*18 + 14*22 + 16*26 + 18*30 + 20*34 + 22*38 = 216 + 308 + 416 + 540 + 680 + 836 = 2996 Σx² = 12² + 14² + 16² + 18² + 20² + 22² = 144 + 196 + 256 + 324 + 400 + 484 = 1804 Σy² = 18² + 22² + 26² + 30² + 34² + 38² = 324 + 484 + 676 + 900 + 1156 + 1444 = 4984 Step 2: Use the Pearson formula: r = [n(Σxy) - (Σx)(Σy)] / sqrt{[n(Σx²) - (Σx)²][n(Σy²) - (Σy)²]} Numerator: n(Σxy) - (Σx)(Σy) = 6(2996) - (102)(168) = 17976 - 17136 = 840 Denominator part 1: n(Σx²) - (Σx)² = 6(1804) - (102)² = 10824 - 10404 = 420 Denominator part 2: n(Σy²) - (Σy)² = 6(4984) - (168)² = 29904 - 28224 = 1680 Product: 420 * 1680 = 705600 Square root: sqrt(705600) = 840 Step 3: r = 840 / 840 = 1.000 Interpretation: r = 1.000 indicates a perfect positive linear relationship. As Matiu's creative writing hours increase, the number of pages he writes increases exactly along a straight line, with no scatter.

2. A research team is studying the relationship between study time and exam scores. They collect data from 15 students and calculate the correlation coefficient as r = 0.72. The team wants to test if this correlation is statistically significant at α = 0.05. Using the critical value table for correlation coefficients with 13 degrees of freedom, the critical value is 0.514. Should the researchers reject the null hypothesis that there is no correlation between study time and exam scores? Answer: A. yes Solution: State the null and alternative hypotheses. The null hypothesis (H0) is that there is no correlation between study time and exam scores in the population, meaning the population correlation coefficient ρ = 0.
   Full step-by-step solution

   Step 1: State the null and alternative hypotheses. The null hypothesis (H0) is that there is no correlation between study time and exam scores in the population, meaning the population correlation coefficient ρ = 0. The alternative hypothesis (H1) is that there is a correlation, so ρ ≠ 0. Step 2: Identify the given values. Sample correlation coefficient r = 0.72 Degrees of freedom (df) = 13 Significance level α = 0.05 Critical value from table = 0.514 Step 3: Understand the decision rule for a two-tailed test. Since α = 0.05 and the test is two-tailed, we compare the absolute value of r to the critical value. If |r| > critical value, we reject H0. If |r| ≤ critical value, we fail to reject H0. Step 4: Compare r to the critical value. |r| = |0.72| = 0.72 Critical value = 0.514 0.72 > 0.514 Step 5: Make the decision. Since 0.72 is greater than 0.514, we reject the null hypothesis. Step 6: Conclusion. The sample correlation of 0.72 is statistically significant at α = 0.05 with 13 degrees of freedom. Therefore, the researchers should reject the null hypothesis that there is no correlation between study time and exam scores. Answer: yes

3. A scatter plot shows the relationship between the number of hours Noah spends on self-study per week (x) and his score on a physics quiz (y) over 6 weeks. The data points are: (1, 31), (2, 36), (4, 46), (5, 51), (7, 61), (8, 66). Calculate the Pearson correlation coefficient r to three decimal places, and interpret its strength and direction. Answer: 1.000 Solution: Compute the sums. n = 6.
   Full step-by-step solution

   Step 1: Compute the sums. n = 6. Σx = 1 + 2 + 4 + 5 + 7 + 8 = 27 Σy = 31 + 36 + 46 + 51 + 61 + 66 = 291 Σxy = 1*31 + 2*36 + 4*46 + 5*51 + 7*61 + 8*66 = 31 + 72 + 184 + 255 + 427 + 528 = 1497 Σx² = 1² + 2² + 4² + 5² + 7² + 8² = 1 + 4 + 16 + 25 + 49 + 64 = 159 Σy² = 31² + 36² + 46² + 51² + 61² + 66² = 961 + 1296 + 2116 + 2601 + 3721 + 4356 = 15051 Step 2: Use the Pearson formula: r = [n(Σxy) - (Σx)(Σy)] / sqrt{[n(Σx²) - (Σx)²][n(Σy²) - (Σy)²]} Numerator: n(Σxy) - (Σx)(Σy) = 6(1497) - (27)(291) = 8982 - 7857 = 1125 Denominator part 1: n(Σx²) - (Σx)² = 6(159) - (27)² = 954 - 729 = 225 Denominator part 2: n(Σy²) - (Σy)² = 6(15051) - (291)² = 90306 - 84681 = 5625 Product: 225 * 5625 = 1265625 Square root: sqrt(1265625) = 1125 Step 3: r = 1125 / 1125 = 1.000 Interpretation: r = 1.000 indicates a perfect positive linear relationship. As Noah's self-study hours increase, his physics quiz score increases exactly along a straight line, with no scatter.

4. Aroha, a plant physiologist, is studying the relationship between the concentration of a specific growth hormone (in mg/L) applied to tomato plants and the average fruit mass (in grams) produced per plant. She collects data from 9 plants and computes the following summary statistics: the mean hormone concentration is 12 mg/L, the mean fruit mass is 150 g, the standard deviation of hormone concentration is 4 mg/L, the standard deviation of fruit mass is 20 g, and the sum of the products of the z-scores (Σzₓzᵧ) is 6.4. Compute the Pearson correlation coefficient r, and interpret the strength and direction of the linear relationship between growth hormone concentration and average fruit mass. Answer: r = 0.80, indicating a strong positive linear relationship between growth hormone concentration and average fruit mass. Solution: Recall the formula for Pearson correlation coefficient using the sum of products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 6.4 and n = 9. Substitute the values: r = 6.4 / (9 - 1) = 6.4 / 8.
   Full step-by-step solution

   Step 1: Recall the formula for Pearson correlation coefficient using the sum of products of z-scores: r = (Σzₓzᵧ) / (n - 1). Here, Σzₓzᵧ = 6.4 and n = 9. Step 2: Substitute the values: r = 6.4 / (9 - 1) = 6.4 / 8. Step 3: Calculate: 6.4 ÷ 8 = 0.80. Step 4: Interpret direction: r = 0.80 is positive, so as hormone concentration increases, fruit mass tends to increase (positive linear relationship). Step 5: Interpret strength: |r| = 0.80 is greater than 0.7, so the relationship is strong. Final answer: r = 0.80, indicating a strong positive linear relationship between growth hormone concentration and average fruit mass.

5. Matiu records the temperature (in degrees Celsius) and the number of ice creams sold at his beachside stall over 7 days. The data points are plotted on a scatter plot: (12, 15), (14, 20), (16, 30), (18, 40), (20, 45), (22, 55), (24, 60). Calculate the Pearson correlation coefficient r for this data, and interpret its strength and direction of the linear relationship. Answer: 0.994 Solution: Compute the sums. n = 7.
   Full step-by-step solution

   Step 1: Compute the sums. n = 7. Σx = 12 + 14 + 16 + 18 + 20 + 22 + 24 = 126 Σy = 15 + 20 + 30 + 40 + 45 + 55 + 60 = 265 Σxy = (12*15) + (14*20) + (16*30) + (18*40) + (20*45) + (22*55) + (24*60) Σxy = 180 + 280 + 480 + 720 + 900 + 1210 + 1440 = 5210 Σx² = 12² + 14² + 16² + 18² + 20² + 22² + 24² Σx² = 144 + 196 + 256 + 324 + 400 + 484 + 576 = 2380 Σy² = 15² + 20² + 30² + 40² + 45² + 55² + 60² Σy² = 225 + 400 + 900 + 1600 + 2025 + 3025 + 3600 = 11775 Step 2: Use the Pearson formula: r = [n(Σxy) - (Σx)(Σy)] / sqrt{[n(Σx²) - (Σx)²][n(Σy²) - (Σy)²]} Numerator: n(Σxy) - (Σx)(Σy) = 7(5210) - (126)(265) = 36470 - 33390 = 3080 Denominator part 1: n(Σx²) - (Σx)² = 7(2380) - (126)² = 16660 - 15876 = 784 Denominator part 2: n(Σy²) - (Σy)² = 7(11775) - (265)² = 82425 - 70225 = 12200 Product: 784 * 12200 = 9564800 Square root: sqrt(9564800) = 3092.70 (approximately) Step 3: r = 3080 / 3092.70 = 0.9959 (approximately) Rounded to three decimal places: r = 0.994 Interpretation: r = 0.994 indicates a very strong positive linear relationship. As the temperature increases, the number of ice creams sold increases very consistently along a nearly perfect straight line.