Question 1183134
Here's how to analyze the relationship between indebtedness and depression using the provided data:

**A) Descriptive Statistics:**

First, let's calculate the descriptive statistics for both variables:

**Indebtedness (Predictor):**

* Mean: (75438 + 89653 + 112653 + 109563 + 56863) / 5 = $88,834
* Median: $89,653 (middle value when sorted)
* Standard Deviation:  Calculate the variance first:
    1. Find the squared differences from the mean for each value.
    2. Sum the squared differences.
    3. Divide by (n-1) where n is the number of data points.
    4. Take the square root of the result.
    Standard Deviation ≈ $21,123.67
* Range: $112,653 - $56,863 = $55,790

**GDS Score (Outcome):**

* Mean: (8 + 9 + 11 + 10 + 6) / 5 = 8.8
* Median: 9
* Standard Deviation: Calculate as above.
    Standard Deviation ≈ 1.92
* Range: 11 - 6 = 5

**B) Correlation Test:**

We'll use Pearson's correlation coefficient (r) to measure the linear association between indebtedness and GDS score.

1. **Calculate the covariance:**

Covariance measures how much the two variables vary together.

Covariance = Σ[(Indebtednessᵢ - Mean of Indebtedness) * (GDSᵢ - Mean of GDS)] / (n - 1)

Covariance ≈ [(75438-88834)(8-8.8) + (89653-88834)(9-8.8) + (112653-88834)(11-8.8) + (109563-88834)(10-8.8) + (56863-88834)(6-8.8)] / 4

Covariance ≈ 34,709.75

2. **Calculate Pearson's correlation coefficient (r):**

r = Covariance / (Standard Deviation of Indebtedness * Standard Deviation of GDS)

r ≈ 34709.75 / (21123.67 * 1.92)

r ≈ 0.85

3. **Hypothesis Test:**

* **Null Hypothesis (H0):** There is no correlation between indebtedness and GDS score (r = 0).
* **Alternative Hypothesis (H1):** There is a correlation between indebtedness and GDS score (r ≠ 0).  This is a two-tailed test.

* **Test Statistic (t):**
    t = r√(n-2) / √(1-r²)
    t = 0.85√(5-2) / √(1-0.85²)
    t ≈ 3.01

* **Degrees of Freedom:** df = n - 2 = 5 - 2 = 3

* **Critical Value:** For α = 0.05 and df = 3 (two-tailed), the critical t-value is approximately ±3.182.

* **P-value:** Using a t-distribution table or calculator, with t=3.01 and df=3, the p-value will be between 0.05 and 0.10.

**C) Interpretation:**

* **Descriptive Statistics:** The descriptive statistics provide a summary of the central tendency and variability of both indebtedness and GDS scores. The positive covariance and correlation coefficient suggest a positive relationship between the two variables.

* **Correlation Test:** The calculated correlation coefficient (r ≈ 0.85) indicates a strong positive linear relationship between indebtedness and GDS score.  However, while the t-statistic (3.01) is close to the critical value (3.182), the p-value is greater than 0.05.

* **Conclusion:** While there appears to be a strong positive correlation, at the α = 0.05 significance level, we fail to reject the null hypothesis.  There is not sufficient statistical evidence from this small sample to conclude that there is a significant correlation between indebtedness and depression as measured by the GDS. This is likely due to the small sample size, which limits the power of the test. A larger sample might yield a statistically significant result.