Question 1178849: You are given the following sample data for variables x and y:
X (independent) | Y (dependent)|
1 | 16 |
7 | 50 |
3 | 22 |
8 | 59 |
11 | 63 |
5 | 46 |
4 | 43 |
1. Construct a scatter plot for these data and describe what, if any, relationship
appears to exist.
2. Calculate product-moment correlation and verify the answer by relating to
scatter plot you derived from the previous question.
3. Compute the regression equation based on these sample data and interpret the
regression coefficients.
4. Based on the sample data, what percentage of the total variation in the dependent
the variable can be explained by the independent variable?
5.Explain why spearman rank correlation coefficient is not suitable to analyze
above data set?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step:
**1. Construct a Scatter Plot and Describe the Relationship:**
* **Scatter Plot:**
* Plot the points (1, 16), (7, 50), (3, 22), (8, 59), (11, 63), (5, 46), and (4, 43) on a graph with x on the horizontal axis and y on the vertical axis.
* You'll observe that as x increases, y tends to increase as well.
* **Relationship:**
* There appears to be a **positive linear relationship** between x and y. As the independent variable (x) increases, the dependent variable (y) also tends to increase. The points roughly follow a straight line pattern.
**2. Calculate Product-Moment Correlation (Pearson's r):**
* **Formula:**
* r = [n(∑xy) - (∑x)(∑y)] / √{[n∑x² - (∑x)²][n∑y² - (∑y)²]}
* **Calculations:**
* n = 7
* ∑x = 1 + 7 + 3 + 8 + 11 + 5 + 4 = 39
* ∑y = 16 + 50 + 22 + 59 + 63 + 46 + 43 = 299
* ∑x² = 1² + 7² + 3² + 8² + 11² + 5² + 4² = 275
* ∑y² = 16² + 50² + 22² + 59² + 63² + 46² + 43² = 14,795
* ∑xy = (1 * 16) + (7 * 50) + (3 * 22) + (8 * 59) + (11 * 63) + (5 * 46) + (4 * 43) = 1,939
* r = [7(1939) - (39)(299)] / √{[7(275) - (39)²][7(14795) - (299)²]}
* r = [13573 - 11661] / √{[1925 - 1521][103565 - 89401]}
* r = 1912 / √[404 * 14164]
* r = 1912 / √5723256
* r = 1912 / 2392.33
* r ≈ 0.7992
* **Verification:**
* The calculated r (0.7992) is positive and relatively close to 1, indicating a strong positive linear relationship. This aligns with the scatter plot, which shows a positive linear trend.
**3. Compute the Regression Equation and Interpret Coefficients:**
* **Regression Equation:** y = a + bx
* **Calculate b (slope):**
* b = [n(∑xy) - (∑x)(∑y)] / [n(∑x²) - (∑x)²]
* b = 1912 / 404
* b ≈ 4.7327
* **Calculate a (y-intercept):**
* a = (∑y / n) - b(∑x / n)
* a = (299 / 7) - 4.7327(39 / 7)
* a ≈ 42.7143 - 26.3379
* a ≈ 16.3764
* **Regression Equation:** y = 16.3764 + 4.7327x
* **Interpretation:**
* **b (4.7327):** For every one-unit increase in x, y is predicted to increase by approximately 4.7327 units.
* **a (16.3764):** When x is 0, the predicted value of y is approximately 16.3764.
**4. Percentage of Total Variation Explained (R²):**
* **R² = r²** (coefficient of determination)
* R² = (0.7992)²
* R² ≈ 0.6387
* **Percentage:** 0.6387 * 100% ≈ 63.87%
* **Interpretation:** Approximately 63.87% of the total variation in y can be explained by the linear relationship with x.
**5. Why Spearman Rank Correlation is Not Suitable:**
* **Spearman Rank Correlation:** This coefficient measures the strength and direction of a monotonic relationship (not necessarily linear) between two ranked variables. It's used when data might not be normally distributed or when relationships are not strictly linear.
* **Suitability for Pearson's r:** The data appears to have a relatively strong linear relationship, as observed in the scatter plot and confirmed by a high Pearson's r value (0.7992).
* **Unnecessary Ranking:** Spearman's rank correlation requires ranking the data. If the relationship is adequately linear, ranking the data loses the magnitude of the differences between the values.
* **Pearson's r is More Powerful:** For linear relationships and normally distributed data (or at least data that don't violate the assumptions of linear regression), Pearson's r is generally considered more powerful and informative than Spearman's rank correlation.
* **Conclusion:** Since the scatter plot suggests a linear relationship, and Pearson's r shows a strong linear correlation, there is no need to rank the data. Pearson's r is more appropriate in this scenario.
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