Variance Spike Investigation for Delivery Times

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Business Context

QuickCart tracks delivery-time consistency because high variability drives support tickets even when average delivery time stays flat. Last month, operations noticed that delivery times in one city looked much more spread out than the historical baseline.

Problem Statement

Decide whether the increase in delivery-time variance is large enough to investigate further using a formal hypothesis test for variance.

Given Data

QuickCart has a long-run historical standard deviation of delivery times of 6 minutes in this city. A random sample of recent deliveries was collected.

Metric	Value
Historical standard deviation	6.0 minutes
Historical variance	36.0 minutes $^2$
Recent sample size	25 deliveries
Recent sample standard deviation	8.0 minutes
Recent sample variance	64.0 minutes $^2$
Significance level	0.05

Assume delivery times are approximately normally distributed, so a chi-square test for one variance is appropriate.

Requirements

State the null and alternative hypotheses for whether variance has increased.
Compute the chi-square test statistic.
Find the critical value or p-value for a one-sided test at $\alpha = 0.05$ .
Decide whether the variance increase is statistically significant.
Briefly explain whether QuickCart should investigate this city further.

Assumptions

The 25 deliveries are a random, independent sample.
The historical variance of 36 minutes $^2$ is the correct baseline.
Delivery times are approximately normal.
The business only cares about an increase in variance, not a decrease.