A city transport authority wants to estimate the mean daily commute time of the city's registered commuters. Every commuter belongs to exactly one of three zones:
| Zone | Commuters |
|---|---|
| Inner | 25,200 |
| Middle | 11,200 |
| Outer | 3,600 |
The authority runs a stratified random sample, treating the three zones as strata. But for staffing reasons each zone office is given the same workload, so the sample is equally allocated: exactly 150 commuters are interviewed in every zone, in total.
The interviews come back with these zone sample means:
| Zone | Sample mean commute (minutes) |
|---|---|
| Inner | 32.4 |
| Middle | 41.0 |
| Outer | 55.0 |
Using this sample, estimate the mean daily commute time for the whole city.
Give your answer in minutes, rounded to 2 decimal places.
Once you get it right, the reasoning behind the answer shows up here.
Explore how values and their corresponding weights combine to pull the average.
() / 0 = 0.0
() / () = 0.00
Pedagogical Insight: Currently Equal: The weighted mean matches the arithmetic mean exactly.
An ordinary arithmetic mean is just a special case of the weighted mean where all weights are equal. When weights are unequal, values with larger weights pull the average more heavily in their direction.