Business Context
FinEdge, a consumer lending company, launched a new underwriting policy and reported that average monthly profit per approved loan increased. Finance leadership wants to know whether that result is statistically credible and how much it depends on modeling assumptions.
Problem Statement
You are given summary results from a 1-month comparison of approved loans before and after the policy change. Determine whether the observed increase in average profit per loan is statistically significant, and explain which assumptions matter most when interpreting the result.
Given Data
| Group | Sample Size | Mean Monthly Profit per Loan | Sample Standard Deviation |
|---|
| Before policy change | 400 | \$118.40 | \$42.70 |
| After policy change | 420 | \$124.90 | \$45.10 |
Additional information:
| Parameter | Value |
|---|
| Significance level | 0.05 |
| Test type | Two-sample, two-tailed |
Requirements
- State the null and alternative hypotheses for the change in mean profit.
- Compute the standard error for the difference in means using Welch's two-sample t-test.
- Calculate the t-statistic.
- Approximate the degrees of freedom and compute the two-tailed p-value.
- Construct a 95% confidence interval for the mean difference.
- Conclude whether the increase is statistically significant at the 5% level.
- Briefly explain which assumptions are required for this conclusion to be reliable and what could go wrong if they are violated.
Assumptions
- The two samples are independent.
- Loans within each period are representative of their populations.
- Profit per loan may be non-normal, but sample sizes are large enough for the sampling distribution of the mean difference to be approximately normal.
- Variances are not assumed equal, so use Welch's t-test.
