SOLUTION: A researcher is interested in the relationship between total student debt after graduating college and depression. In order to test the hypothesis that students with more debt are

Question 1183134: A researcher is interested in the relationship between total student debt after graduating
college and depression. In order to test the hypothesis that students with more debt are more
depressed, the researcher conducts a cross-sectional study that inquires about indebtedness
after graduation and asks participants to complete the geriatric depression scale (GDS) test to
quantify their depression on a scale of 1-15, 15 being the most depressed. The following
table should be considered a SRS of participants’ responses relative to the variables of
interest.
Indebtedness at Graduation ($) Performance on GDS (1-15)
75,438 8
89,653 9
112,653 11
109,563 10
56,863 6
A) Calculate basic descriptive statistics for your predictor and outcome variables.
B) Perform a formal test addressing the correlation between your predictor and outcome
variables. (Use alpha = 0.05).
C) Interpret your results.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
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.

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