Business Context
BrewBox, a subscription coffee company, wants a simple baseline forecast for next week’s order volume before committing inventory. The operations team asked for a quick statistical approach using recent weekly demand.
Problem Statement
Use the last 8 weeks of orders to build a linear trend forecast for week 9, quantify forecast uncertainty, and assess whether the trend is strong enough to rely on for short-term planning.
Given Data
| Week | Orders |
|---|
| 1 | 482 |
| 2 | 495 |
| 3 | 501 |
| 4 | 515 |
| 5 | 528 |
| 6 | 540 |
| 7 | 548 |
| 8 | 561 |
Assume a simple linear model:
Orderst=β0+β1t+εt
Use a 95% confidence level for the trend estimate and a 95% prediction interval for the week 9 forecast.
Requirements
- Estimate the linear regression equation relating week number to orders.
- Compute the forecasted orders for week 9.
- Calculate the residual variance and standard error of the regression.
- Test whether the slope is different from 0 at α=0.05.
- Construct a 95% prediction interval for week 9 orders.
- Briefly explain how you would approach forecasting in practice beyond this simple baseline.
Assumptions
- Weekly demand follows an approximately linear short-term trend.
- Residuals are independent with constant variance.
- No major promotions, stockouts, or holiday shocks affected these 8 weeks.
- This is a baseline forecast, not a full production forecasting system.