A 41.5-Minute Average, a 26% Response Rate and Three Excluded Centres
The Customer Service Division reports quarterly on counter waiting times across 12 service centres. For the latest quarter it reports: 'Average wait 41.5 minutes, well above our 35-minute target.' The figure comes from a random sample of 64 centre-days drawn from the quarter; the sample standard deviation is 16.0 minutes. Separately, a satisfaction survey was mailed to 1,200 customers who visited during the quarter. 312 responded, and 68% of respondents said they were satisfied. The executive director proposes to state in the annual report that '68% of customers are satisfied, within a margin of error of about 3 percentage points'. A board member notes that the three largest centres, which handle 46% of all visits, were excluded from the waiting-time sample because their counters were being renovated during the quarter. The division also reports that average wait improved from 44.0 to 41.5 minutes over the year, but the mix of transactions shifted heavily toward complex cases.
1. Test whether the mean waiting time exceeds the 35-minute target and report the 95% confidence interval. Interpret both.
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Hypotheses
H0: mu = 35 minutes (target met). H1: mu > 35 minutes (target exceeded), one-tailed, because the policy question is whether waits are worse than target. Use alpha = 0.05.
Test statistic
Standard error = s / sqrt(n) = 16.0 / sqrt(64) = 16.0 / 8 = 2.0.
t = (41.5 - 35.0) / 2.0 = 6.5 / 2.0 = 3.25, with df = 63.
The one-tailed critical value at alpha = 0.05 is about 1.67 and at 0.01 about 2.39. t = 3.25 exceeds both; p is roughly 0.001.
Decision: reject H0. The data are not consistent with a true mean of 35 minutes; the evidence is that the population mean wait exceeds target.
95% confidence interval
41.5 +/- 1.998 x 2.0 = 41.5 +/- 4.0, giving 37.5 to 45.5 minutes.
Interpretation: we are 95% confident the true average wait across the quarter is between about 37.5 and 45.5 minutes. The whole interval lies above 35, which is the same conclusion as the test but more informative for management: the shortfall is not marginal. Reporting only '41.5 vs 35' hides the uncertainty; reporting the interval shows the target is missed by between 2.5 and 10.5 minutes on average.
Caveat: 64 centre-days is a modest sample, and the exclusion of the three largest centres (next task) means this interval applies at best to the sampled centres, not to the system as a whole.
2. Is the claim of '68% satisfied, within about 3 percentage points' defensible? Calculate the correct margin and explain what else limits the estimate.
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The arithmetic does not support 3 points.
Sample proportion p = 0.68, n = 312.
Standard error = sqrt(p(1-p)/n) = sqrt(0.68 x 0.32 / 312) = sqrt(0.2176 / 312) = sqrt(0.000697) = 0.0264.
95% margin = 1.96 x 0.0264 = 0.0518, i.e. about +/-5.2 percentage points, giving 62.8% to 73.2%.
With a finite population correction for a population of about 1,200: sqrt((1200 - 312)/(1200 - 1)) = sqrt(888/1199) = 0.861, so the margin becomes 5.18 x 0.861 = about +/-4.5 points. Still not 3 points. A margin of 3 points would need an n of about 930.
The larger problem is non-response. 312 of 1,200 is a 26% response rate. The sampling error above assumes respondents are a random subset of visitors; they are not. People with strong views - and people with time on their hands - are more likely to reply, and the direction of that bias is unknown. Non-response bias is not captured by the margin of error and can easily exceed 5 points.
Recommended wording: 'Of the 312 customers who responded (26% of those surveyed), 68% reported being satisfied (95% confidence interval 63% to 73%). Results may not be representative of all customers.' Better still, weight the responses to known visit characteristics, run a follow-up by phone or SMS to lift the rate, and report the response rate alongside the result. A +/-3 point claim in an annual report would be a misstatement of precision.
3. What is the effect of excluding the three largest centres, and could the reported improvement from 44.0 to 41.5 minutes be an artefact? Explain how you would test the improvement properly.
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Exclusion of the three largest centres is a coverage problem, not a sampling problem. They handle 46% of all visits, so the reported average describes slightly more than half the service. If those centres are the busiest, their queues are likely longer than average, so the reported 41.5 minutes probably understates the system-wide wait. The confidence interval computed earlier is valid for the sampled centre-days only; it says nothing about the excluded centres.
Could the improvement be an artefact? Yes. The reported fall is in the overall mean, and the transaction mix shifted heavily toward complex cases. Complex cases take longer. If the mix moved toward long transactions, the overall mean could fall while every transaction-type mean stayed the same or rose - a Simpson's paradox pattern.
Numerical illustration: suppose last year 70% of visits were simple (20 minutes) and 30% complex (100 minutes): 0.7 x 20 + 0.3 x 100 = 44.0. This year the split is 45%/55%, and if each type got faster, say simple 18 and complex 60: 0.45 x 18 + 0.55 x 60 = 8.1 + 33.0 = 41.1. The overall mean fell while every stratum improved - or, in a worse variant, the overall mean falls while every stratum worsens.
How to test properly
- Stratify by transaction type and compare like with like; report per-stratum means and intervals.
- Weight the centre-day sample by visit volume so the estimate reflects the customer experience, and bring the three excluded centres back in (or state their exclusion as a limitation with an estimate of its size and direction).
- Use centre as a clustering variable; waits within a centre are correlated, so the effective standard error is larger than the simple formula suggests.
- Publish the stratified results and the method alongside the headline figure.