Business Context
SwiftAir wants to know whether last month’s customer complaint rate was unusually high or just normal random variation. The analytics team needs to choose the right statistical concepts to interpret the scenario correctly.
Problem Statement
Historically, 4.0% of completed trips generate a customer complaint. In a recent month, SwiftAir completed 2,500 trips and received 125 complaints. Determine whether the observed complaint rate is significantly different from the historical benchmark, and quantify the uncertainty around the new rate.
Given Data
| Metric | Value |
|---|
| Historical complaint rate | 4.0% |
| Recent month trips | 2,500 |
| Recent month complaints | 125 |
| Observed recent complaint rate | 5.0% |
| Significance level | 0.05 |
| Confidence level | 95% |
Assume the historical 4.0% benchmark is stable and can be treated as the null value.
Requirements
- State the null and alternative hypotheses for whether the recent complaint rate differs from the historical rate.
- Identify the appropriate distributional approximation and justify why it is reasonable.
- Compute the standard error under the null hypothesis.
- Calculate the z-statistic and two-sided p-value.
- Construct a 95% confidence interval for the recent month complaint rate.
- Conclude whether the recent month should be treated as statistically unusual.
- Briefly explain which statistical concepts are being used and why they are appropriate for this scenario.
Assumptions
- Each trip independently results in either a complaint or no complaint.
- The sample of 2,500 trips is representative of normal operations.
- The normal approximation to the binomial is acceptable because expected successes and failures are both sufficiently large.
