Business Context

ColdBox, a meal-kit subscription company, needs a short-term forecast of weekly order volume to plan packaging and delivery capacity. The operations team wants a simple time-series forecast using recent demand data.

Problem Statement

You are given 8 weeks of historical orders. Build a simple linear time-series trend model using week number as the predictor, then forecast demand for week 9. Also quantify forecast uncertainty.

Given Data

Assume the model:

$Y_t = a + bt + \varepsilon_t$

where $Y_t$ is weekly orders, $t$ is week number, and errors are independent with mean 0 and constant variance.

Use a 95% confidence level. For this question, use the residual standard error from the fitted model to estimate forecast uncertainty.

Requirements

1. Explain briefly what time-series forecasting is in practical terms.
2. Fit a linear trend model and calculate the slope and intercept.
3. Forecast orders for week 9.
4. Compute residuals, the residual standard error, and a 95% prediction interval for week 9.
5. Interpret whether the forecast is reliable enough for staffing and packaging decisions.

Assumptions

• No strong seasonality is present in these 8 weeks.
• A linear trend is a reasonable first approximation.
• Forecast uncertainty is based on historical residual variation.
• Use the t-distribution for the prediction interval with $n-2$ degrees of freedom.