The Physiotherapy Slot Experiment
Greenfield Physiotherapy has two physiotherapists, each rostered for 7.5 clinical hours a day, five days a week. Appointments are currently 30 minutes, and the clinic books 28 slots per day. Last quarter, 19% of booked appointments were not attended (DNA), and each DNA leaves the slot empty. The practice manager proposes moving to 20-minute appointments, which would allow 42 slots per day, but expects DNA to rise to 24% because shorter appointments are easier to miss. A neighbouring clinic reports a 'mean DNA rate of 16%' across its three sites. Its figures are 8% at a 60-slot city site, 9% at a 40-slot site, and 31% at a 12-slot rural site. The practice manager cites the 16% as evidence that shorter appointments do not increase non-attendance. The clinic receives £42 per attended appointment. Each physiotherapist costs £115 per day, and fixed overhead is £180 per day.
1. Calculate the number of attended appointments per day under each model, using the stated assumptions. Show your arithmetic and state which model produces more attended appointments.
Read the worked answer
Current model: 28 booked slots × (1 − 0.19) = 28 × 0.81 = 22.68 attended appointments per day. New model: 42 booked slots × (1 − 0.24) = 42 × 0.76 = 31.92 attended appointments per day. The difference is 31.92 − 22.68 = 9.24 attended appointments per day, or about 46 per five-day week.
The new model produces more attended appointments despite the higher DNA rate, because the increase in offered slots (42/28 = 1.5, a 50% increase) more than offsets the rise in non-attendance (0.81 to 0.76, a 6.2% relative fall in attendance per slot). The conclusion depends on demand being sufficient to fill 42 slots and on the 24% DNA estimate being accurate. If DNA rose above 46%, the advantage would disappear: solve 42(1 − d) = 22.68, giving d = 0.46.
2. The practice manager says the neighbouring clinic's mean DNA rate of 16% shows that shorter appointments do not increase non-attendance. Evaluate this claim.
Read the worked answer
The 16% is an unweighted mean of 8%, 9% and 31%: (8 + 9 + 31)/3 = 16%. That gives the 12-slot rural site the same weight as the 60-slot city site, even though the city site has five times as many slots. Weighted by slots, the rate is (0.08×60 + 0.09×40 + 0.31×12)/112 = (4.8 + 3.6 + 3.72)/112 = 12.12/112 = 10.8%. So the headline 16% is inflated by a small outlier site and is not a good summary of the neighbour's overall DNA rate.
More importantly, the comparison cannot isolate appointment length. The three sites differ in setting, list size and patient population, and we are not told the appointment length at any of them. The 31% rural site may have long travel times or an older population. A mean across heterogeneous sites is a confounded comparison, not evidence that shorter appointments are safe. The manager would need appointment-level data with length and DNA recorded together, ideally within the same site.
3. The clinic earns £42 per attended appointment and pays each physiotherapist £115 per day. Fixed overhead is £180 per day. Which model gives the higher daily contribution, and by how much?
Read the worked answer
Daily staff cost is 2 × £115 = £230; adding £180 overhead gives £410 per day under either model.
Current model: 22.68 attended × £42 = £952.56 revenue. Contribution = £952.56 − £410 = £542.56.
New model: 31.92 attended × £42 = £1,340.64 revenue. Contribution = £1,340.64 − £410 = £930.64.
Difference = £930.64 − £542.56 = £388.08 per day, or £1,940.40 over a five-day week. The new model is better on these assumptions. The conclusion is sensitive to the DNA estimate: at 24% DNA the new model wins clearly, but if DNA rose above 46% it would produce fewer attended appointments than the current model and the contribution advantage would disappear. It also assumes demand fills the extra slots and that no additional reception or admin cost is created.