Math Problem Statement

While making calculations about the correlation coefficient, a student obtained the following results: n = 25, ΣX = 125, ΣX² = 650, ΣY = 100, ΣY² = 460, ΣXY = 508. Later, it was discovered that two pairs of values were recorded incorrectly as X = 6, Y = 12 and X = 8, Y = 8, while the correct values were X = 6, Y = 14 and X = 8, Y = 6. Calculate the corrected correlation coefficient.

Solution

The problem asks us to calculate the corrected correlation coefficient based on the given statistics after rectifying some incorrect data points. Let's go through the solution step by step:

Given Data:

  • n=25n = 25
  • ΣX=125\Sigma X = 125
  • ΣX2=650\Sigma X^2 = 650
  • ΣY=100\Sigma Y = 100
  • ΣY2=460\Sigma Y^2 = 460
  • ΣXY=508\Sigma XY = 508

Incorrect Data Points:

  • (X1,Y1)=(6,14)(X_1, Y_1) = (6, 14), incorrectly recorded as (6,12)(6, 12)
  • (X2,Y2)=(8,6)(X_2, Y_2) = (8, 6), incorrectly recorded as (8,8)(8, 8)

Correcting the Summations:

For each incorrect data point, update the summations accordingly.

1. Update ΣX\Sigma X:

No change, since XX values were correctly recorded.

ΣXcorrected=125\Sigma X_{\text{corrected}} = 125

2. Update ΣY\Sigma Y:

Replace the incorrect YY-values with the correct ones:

= 100 - 20 + 20 = 100$$ No change in $$\Sigma Y$$. #### 3. Update $$\Sigma XY$$: For the products $$XY$$, replace the incorrect pairs with the correct ones: $$\Sigma XY_{\text{corrected}} = \Sigma XY - (6 \times 12 + 8 \times 8) + (6 \times 14 + 8 \times 6)$$ $$\Sigma XY_{\text{corrected}} = 508 - (72 + 64) + (84 + 48)$$ $$\Sigma XY_{\text{corrected}} = 508 - 136 + 132 = 504$$ #### 4. Update $$\Sigma Y^2$$: Replace the incorrect $$Y^2$$-values with the correct ones: $$\Sigma Y^2_{\text{corrected}} = \Sigma Y^2 - (12^2 + 8^2) + (14^2 + 6^2)$$ $$\Sigma Y^2_{\text{corrected}} = 460 - (144 + 64) + (196 + 36)$$ $$\Sigma Y^2_{\text{corrected}} = 460 - 208 + 232 = 484$$ ### Corrected Correlation Coefficient Formula: The formula for the Pearson correlation coefficient is: $$r = \frac{n \Sigma XY - \Sigma X \Sigma Y}{\sqrt{\left(n \Sigma X^2 - (\Sigma X)^2\right) \left(n \Sigma Y^2 - (\Sigma Y)^2\right)}}$$ Substitute the corrected values: - $$n = 25$$ - $$\Sigma X = 125$$ - $$\Sigma Y = 100$$ - $$\Sigma X^2 = 650$$ - $$\Sigma Y^2 = 484$$ - $$\Sigma XY = 504$$ #### Numerator: $$n \Sigma XY - \Sigma X \Sigma Y = 25(504) - (125)(100)$$ $$= 12600 - 12500 = 100$$ #### Denominator: $$\sqrt{\left(n \Sigma X^2 - (\Sigma X)^2\right) \left(n \Sigma Y^2 - (\Sigma Y)^2\right)}$$ Calculate each term separately: - $$n \Sigma X^2 - (\Sigma X)^2 = 25(650) - 125^2$$ $$= 16250 - 15625 = 625$$ - $$n \Sigma Y^2 - (\Sigma Y)^2 = 25(484) - 100^2$$ $$= 12100 - 10000 = 2100$$ So the denominator becomes: $$\sqrt{625 \times 2100} = \sqrt{1312500} = 1145.64$$ #### Final Calculation: $$r = \frac{100}{1145.64} \approx 0.0873$$ ### Final Answer: The corrected correlation coefficient is approximately: $$r \approx 0.087$$ Let me know if you would like further clarifications! --- ### Related Questions: 1. How is the Pearson correlation coefficient affected by data errors? 2. Can rounding during corrections impact correlation results significantly? 3. What are alternative methods to compute correlation if the dataset is very large? 4. How does the sample size $$n$$ influence the reliability of $$r$$? 5. What does an $$r$$ value close to 0 imply about the relationship between $$X$$ and $$Y$$? ### Tip: Always double-check your data before computing statistics to avoid unnecessary recalculations!

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation Coefficient
Summation Corrections

Formulas

Pearson correlation coefficient formula: r = (nΣXY - ΣXΣY) / sqrt((nΣX² - ΣX²)(nΣY² - ΣY²))

Theorems

-

Suitable Grade Level

Grades 11-12 and Undergraduate

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