Business Context
PulseDesk, a B2B customer support platform, is reviewing weekly ticket volume after a pricing change. The operations lead wants to know whether the recent pattern reflects normal week-to-week variability or a real upward trend that requires staffing changes.
Problem Statement
Use the 8 weeks of ticket data below to distinguish variance analysis from trend analysis. Specifically, quantify overall variability around the mean and test whether there is evidence of a positive linear trend over time.
Given Data
| Week | Tickets |
|---|
| 1 | 482 |
| 2 | 495 |
| 3 | 501 |
| 4 | 509 |
| 5 | 515 |
| 6 | 528 |
| 7 | 536 |
| 8 | 548 |
Additional parameters:
| Parameter | Value |
|---|
| Significance level | 0.05 |
| Number of weeks | 8 |
Requirements
- Compute the sample mean and sample variance of weekly tickets.
- Explain what variance analysis tells you in this setting.
- Fit a simple linear trend model: yt=a+bt, where t is week number.
- Estimate the slope b and test whether the trend is significantly positive.
- Explain what trend analysis tells you that variance analysis does not.
- Give a business recommendation on whether staffing should change.
Assumptions
- Weekly ticket counts are measured consistently across all 8 weeks.
- A linear trend is a reasonable first-pass model over this short period.
- Residuals from the trend model are treated as approximately independent for this interview exercise.
- Use a one-sided test for whether the slope is greater than 0.
