Homework 5: Exact and Approximate Inference for Call Center Waiting Times

Background

Waiting times in service systems are often modeled using the exponential distribution.

Suppose

\[ X_1,\dots,X_n \sim \Exp(\theta), \]

where \(\theta\) is the mean waiting time.

A key property of the exponential distribution is that

\[ \sum_{i=1}^n X_i \sim \Gam(n,\theta). \]

Equivalently,

\[ \frac{2\sum_{i=1}^n X_i}{\theta} \sim \chi^2_{2n}. \]

This allows exact inference for \(\theta\).

Data

The context for this assignment is based on real data from the New York City Open Data portal:

NYC Open Data. 311 Service Requests from 2010 to Present.
The observations used here are inter-arrival waiting times (in seconds) computed from service-request timestamps for a fixed complaint type.

For this homework, we use a subset of 20 waiting times selected from that real dataset.

The following waiting times (in seconds) were observed:

4 7 8 11 29 34 36 41 47 49
53 65 79 89 134 134 165 178 227 375

Thus,

\[ n = 20. \]

Question

Suppose that a service target states that the mean waiting time should not exceed 60 seconds.

We test

\[ H_0:\theta = 60 \qquad\text{vs}\qquad H_a:\theta > 60. \]

Use significance level

\[ \alpha = 0.05. \]

Part 1: Model Discussion

Explain why the exponential distribution is often used to model waiting times.

Discuss at least two properties that make it reasonable in this context.

Part 2: Exact Hypothesis Test

Using the result above:

  1. Derive the rejection region for a level-\(\alpha\) test of \(H_0:\theta=\theta_0 \text{ vs } H_a:\theta>\theta_0\).

  2. Apply the test to the dataset above with \(\theta_0 = 60\).

  3. Compute the p-value.

  4. State your conclusion in context.

Part 4: Exact Upper Confidence Bound

Derive a \((1-\alpha)\) upper confidence bound for \(\theta\) of the form \([0,\theta_U]\). Compute this bound for the dataset and interpret it in context.

Part 5: Normal Approximation

Using the Central Limit Theorem, construct the approximate test statistic in terms of the sample mean, \(\barX\).

  1. Compute the approximate p-value.

  2. Construct the approximate 95% upper confidence bound.

Part 6: Comparison and Discussion

Compare the exact exponential inference and the normal approximation.

Discuss:

  1. the difference between the two p-values,
  2. the difference between the two upper confidence bounds,
  3. whether the two methods lead to the same practical conclusion,
  4. how skewness affects the approximation.

Submission

Your work should include:

  • clear derivations,
  • numerical calculations,
  • interpretation in context,
  • sufficient detail for reproducibility.