Cold-Brew Trial Rate: Reweighting, Bias, and Sampling Error
A regional café chain is deciding whether to keep a new cold-brew drink on the menu. Staff surveyed 200 customers at a single branch on Tuesday and Wednesday afternoons between 2 and 5 pm, asking whether they had tried the drink. Of the 110 respondents under 30, 44 had tried it; of the 90 respondents aged 30 and over, 30 had tried it. Chain loyalty data show that 45% of customers across all branches are under 30, and the surveyed branch has a similar age profile. The drink would sell for $4.75 with $2.65 of ingredient cost, and the branch serves about 900 customers a day. The marketing lead wants one headline trial-rate number to plug into a revenue forecast; the operations lead wants to know whether the survey can be used as it stands.
1. Compute the trial rate within each age group and overall. Identify which single number is most misleading as a headline and why.
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Conditional (within-group) trial rates:
- Under 30: 44/110 = 0.400 (40.0%)
- 30 and over: 30/90 = 0.3333 (33.3%)
Overall: (44 + 30)/200 = 74/200 = 0.370 (37.0%).
Which is misleading: the 37.0% overall figure, if presented as a chain-wide rate. It is a weighted average of 40.0% and 33.3%, but with weights 55%/45% (110 of 200 under 30) rather than the chain's true 45%/55%. Because the higher-trial group is over-represented, 37.0% sits above the correctly weighted estimate, and using it as the headline slightly inflates the forecast. The 40.0% and 33.3% group rates are themselves unbiased within their groups, but they are still subject to a separate problem: the sample is drawn from one branch, on weekday afternoons, which is not representative of the chain's customer mix by daypart.
2. Re-weight the group rates to the chain's age mix and convert the result into a daily gross-margin forecast. Quantify how much the re-weighting changes the answer.
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Re-weighted trial rate: 0.45 × 0.400 + 0.55 × 0.3333 = 0.1800 + 0.1833 = 0.3633 (36.3%).
Daily forecast (900 customers):
- Re-weighted: 0.3633 × 900 ≈ 327 trial customers
- Unadjusted: 0.370 × 900 = 333 trial customers
- Difference: about 6 customers per day.
Margin per unit: $4.75 − $2.65 = $2.10.
- Re-weighted gross margin: 327 × $2.10 = $686.70/day
- Unadjusted: 333 × $2.10 = $699.30/day
- Gap: about $12.60/day, or roughly $4,600/year.
Judgement: re-weighting is the right thing to do, but it moves the forecast by only 0.7 percentage points. The larger risk sits outside the arithmetic: a customer who tried the drink on a promotional afternoon is not the same as a repeat buyer, so a trial rate is an upper bound on demand, not a revenue estimate. The forecast should be labelled 'trial demand' and discounted for repeat conversion before anyone commits to a menu decision.
3. Compute the standard error of the overall proportion and explain what it implies about the credibility of the 37.0% versus 36.3% difference and about the survey's main weakness.
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Standard error of the overall proportion (p̂ = 0.37, n = 200): SE = √(p̂(1 − p̂)/n) = √(0.37 × 0.63 / 200) = √(0.2331/200) = √0.0011655 ≈ 0.0341, about 3.4 percentage points.
Approximate 95% margin of error: 2 × 3.4 ≈ 6.8 points, so the plausible range is roughly 30% to 44%.
Implication for 37.0% vs 36.3%: the 0.7-point re-weighting shift is only about one-fifth of one standard error - far inside sampling noise. Reporting either figure to a decimal place is false precision; both should be quoted as 'about 36–37%, plus or minus 7 points'.
Main weakness: the dominant threat is bias, not variance. Weekday 2–5 pm sampling at one branch systematically under-covers commuters, evening and weekend customers, and other neighbourhoods. Bias of that kind does not shrink as n grows - surveying 2,000 afternoon customers would produce a very precise estimate of the wrong population. The defensible next step is a stratified sample across dayparts and at least three branches, with the age weights fixed in advance, before any number goes into the forecast.