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.

For exponential data, an important fact is that

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

This gives an exact method for inference about \(\theta\).

We can also use a Normal approximation based on the Central Limit Theorem.

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: Looking at the Data

  1. Compute the sample mean \(\barX\).

  2. Make a simple plot of the data, such as a histogram, dotplot, or stem-and-leaf display.

  3. Comment briefly on whether the data appear right-skewed.

  4. Explain briefly why an exponential distribution might be a reasonable model for waiting times.

Part 2: Exact Hypothesis Test

For the exact test, use

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

  1. Compute the test statistic for testing \(H_0:\theta=60\).

  2. Compute the p-value.

  3. State your conclusion in context.

Part 3: Exact Upper Confidence Bound

Using the exponential model, construct a 95% upper confidence bound for \(\theta\) of the form

\[ [0,\theta_U]. \]

Compute this bound and interpret it in context.

Part 4: Normal Approximation

Using the Central Limit Theorem, approximate the distribution of \(\barX\) by a Normal distribution and construct the corresponding test statistic.

  1. Compute the approximate p-value.

  2. Construct the approximate 95% upper confidence bound.

Part 5: Comparison and Discussion

Compare the exact exponential inference and the Normal approximation.

Discuss:

  1. the two p-values,

  2. the two upper confidence bounds,

  3. whether the two methods lead to the same practical conclusion,

  4. whether the Normal approximation seems reasonable here.

Submission

Your work should include:

  • numerical calculations,
  • a plot of the data,
  • clear conclusions in context,
  • and enough explanation that another student could follow your reasoning.