Calculate the Pearson product-moment correlation coefficient (r) and find the variance and standard deviation…

Question

Calculate the Pearson product-moment correlation coefficient (r) and find the variance and standard deviation for the given datasets.

ScanSolve AI Tutor · Accepted Answer

1. Calculate the Pearson product-moment correlation coefficient (r) Step 1: Create a table to calculate ∑ X, ∑ Y, ∑ XY, ∑ X², and ∑ Y². Given data: English (X): 1, 2, 3, 4, 5 Mathematics (Y): 2, 4, 5, 4, 5 Number of students, n = 5. c — c — c — c — c X & Y & XY & X² & Y² 1 & 2 & 2 & 1 & 4 2 & 4 & 8 & 4 & 16 3 & 5 & 15 & 9 & 25 4 & 4 & 16 & 16 & 16 5 & 5 & 25 & 25 & 25 ∑ X = 15 & ∑ Y = 20 & ∑ XY = 66 & ∑ X² = 55 & ∑ Y² = 86 Step 2: Apply the Pearson correlation coefficient formula. The formula for Pearson product-moment correlation coefficient (r) is: r = (n ∑ XY − (∑ X)(∑ Y))/(√([n ∑ X² − (∑ X)²][n ∑ Y² − (∑ Y)²])) Substitute the calculated values: r = (5(66) − (15)(20))/(√([5(55) − (15)²][5(86) − (20)²])) Step 3: Simplify the expression. r = (330 − 300)/(√([275 − 225][430 − 400])); r = (30)/(√([50][30])); r = (30)/(√(1500)); r = 30/38.7298 r ≈ 0.7746 The Pearson product-moment correlation coefficient is 0.7746. 2. Find the variance and standard deviation of the following set of data: 5, 8, 12, 15 Step 1: Calculate the mean (X) of the data. Given data: X = 5, 8, 12, 15; Number of data points, n = 4. ∑ X = 5 + 8 + 12 + 15 = 40; X = (∑ X)/(n) = 40/4 = 10 Step 2: Calculate the sum of squared differences from the mean, ∑ (X − X)². c — c — c X & X − X & (X − X)² 5 & 5 − 10 = −5 & 25; 8 & 8 − 10 = −2 & 4; 12 & 12 − 10 = 2 & 4; 15 & 15 − 10 = 5 & 25; ∑ (X − X)² = 58 Step 3: Calculate the variance (σ²). Assuming this is a population, the variance formula is: σ² = ∑ (X − X)²n; σ² = 58/4 = 14.5 Step 4: Calculate the standard deviation (σ). The standard deviation is the square root of the variance: σ = √(σ²) = √(14.5) ≈ 3.8079 The variance is 14.5 and the standard deviation is 3.8079. 4. Using Spearman's rank, obtain the correlation coefficient of the data set below. Step 1: Rank the X (Theory) and Y (Practical) values separately. Given data: Theory (X): 8, 3, 9, 2, 7, 10, 4, 6, 1, 5 Practical (Y): 9, 5, 10, 1, 8, 7, 3, 4, 2, 6 Number of data pairs, n = 10. Step 2: Create a table to calculate the ranks, differences in ranks (d), and squared differences (d²). c — c — c — c — c — c X & Y & Rank X (Rx) & Rank Y (Ry) & d = Rx − Ry & d² 8 & 9 & 8 & 9 & −1 & 1 3 & 5 & 3 & 5 & −2 & 4 9 & 10 & 9 & 10 & −1 & 1 2 & 1 & 2 & 1 & 1 & 1 7 & 8 & 7 & 8 & −1 & 1 10 & 7 & 10 & 7 & 3 & 9 4 & 3 & 4 & 3 & 1 & 1 6 & 4 & 6 & 4 & 2 & 4 1 & 2 & 1 & 2 & −1 & 1 5 & 6 & 5 & 6 & −1 & 1 ∑ d² = 24 Step 3: Apply Spearman's rank correlation coefficient formula. The formula for Spearman's rank correlation coefficient (r_s) is: r_s = 1 − (6 ∑ d²)/(n(n² − 1)); Substitute n = 10 and ∑ d² = 24: r_s = 1 − (6(24))/(10(10² − 1)); r_s = 1 − (144)/(10(100 − 1)); r_s = 1 − (144)/(10(99)); r_s = 1 − 144/990; r_s = 1 − 0.145454... r_s ≈ 0.8545 The Spearman's rank correlation coefficient is 0.8545.

Calculate the Pearson product-moment correlation coefficient (r) and find the variance and standard deviation for the given datasets.

Step-by-step solution

Create a table to calculate $\sum X$ , $\sum Y$ , $\sum XY$ , $\sum X^2$ , and $\sum Y^2$ .

Apply the Pearson correlation coefficient formula.

Simplify the expression.

Calculate the mean ( $\bar{X}$ ) of the data.

Calculate the sum of squared differences from the mean, $\sum (X - \bar{X})^2$ .

Calculate the variance ( $\sigma^2$ ).

Calculate the standard deviation ( $\sigma$ ).

Rank the X (Theory) and Y (Practical) values separately.

Create a table to calculate the ranks, differences in ranks ( $d$ ), and squared differences ( $d^2$ ).

Apply Spearman's rank correlation coefficient formula.

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More Mathematics Questions

Calculate the Pearson product-moment correlation coefficient (r) and find the variance and standard deviation for the given datasets.

Step-by-step solution

Create a table to calculate ∑X\sum X∑X, ∑Y\sum Y∑Y, ∑XY\sum XY∑XY, ∑X2\sum X^2∑X2, and ∑Y2\sum Y^2∑Y2.

Apply the Pearson correlation coefficient formula.

Simplify the expression.

Calculate the mean (Xˉ\bar{X}Xˉ) of the data.

Calculate the sum of squared differences from the mean, ∑(X−Xˉ)2\sum (X - \bar{X})^2∑(X−Xˉ)2.

Calculate the variance (σ2\sigma^2σ2).

Calculate the standard deviation (σ\sigmaσ).

Rank the X (Theory) and Y (Practical) values separately.

Create a table to calculate the ranks, differences in ranks (ddd), and squared differences (d2d^2d2).

Apply Spearman's rank correlation coefficient formula.

Still stuck on this one?

More Mathematics Questions

Create a table to calculate $\sum X$ , $\sum Y$ , $\sum XY$ , $\sum X^2$ , and $\sum Y^2$ .

Calculate the mean ( $\bar{X}$ ) of the data.

Calculate the sum of squared differences from the mean, $\sum (X - \bar{X})^2$ .

Calculate the variance ( $\sigma^2$ ).

Calculate the standard deviation ( $\sigma$ ).

Create a table to calculate the ranks, differences in ranks ( $d$ ), and squared differences ( $d^2$ ).