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

Assume a simple linear model:

$Orders_t = \beta_0 + \beta_1 t + \varepsilon_t$

Use a 95% confidence level for the trend estimate and a 95% prediction interval for the week 9 forecast.

Requirements

1. Estimate the linear regression equation relating week number to orders.
2. Compute the forecasted orders for week 9.
3. Calculate the residual variance and standard error of the regression.
4. Test whether the slope is different from 0 at $\alpha = 0.05$.
5. Construct a 95% prediction interval for week 9 orders.
6. 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.