AP Statistics · Unit 9 · Inference for Quantitative Data: Slopes

A coffee roastery's quality-control manager wants to understand how roast duration affects the moisture percentage remaining in roasted beans. She randomly selected $n=16$ batches of the same bean variety from routine production, where each batch had already been scheduled for a particular roast duration (ranging from 8 to 20 minutes) as part of normal operations; roast duration was not randomly assigned by the manager. For each batch she recorded the roast duration, in minutes, and the moisture percentage remaining in the beans after roasting. A residual plot for the least-squares regression showed a random scatter of points with no leftover curved pattern and a roughly constant vertical spread across all durations. A normal probability plot of the residuals was approximately linear, with no strong skew or outliers. The 16 sampled batches represent well under 10% of all batches the roastery produces in a year. Computer output from the regression analysis is shown below.

$S = 1.15$, $R\text{-sq} = 62.8\%$

(a) The least-squares regression equation is $\widehat{\text{Moisture}} = 18.42 - 0.417(\text{Duration})$, where Duration is roast time in minutes and Moisture is the percentage of moisture remaining in the beans after roasting. Interpret the slope of this regression line in context.
(b) Construct and interpret a 95% confidence interval for the slope of the population regression line relating moisture percentage to roast duration. Use $t^{*} = 2.145$ ($df = 14$).
(c) Does the interval from part (b) provide convincing statistical evidence that increasing the roast duration causes a decrease in the moisture percentage remaining in the beans? Justify your answer.