Question 1178849
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.