Estimates, standard errors, and confidence intervals
Use the uncertainty around an average packing time or a sales conversion rate to decide whether the evidence supports a commitment.
Interpret a confidence interval for a mean or conversion rate without treating it as a guarantee about one future outcome.
How much packing time should the manager budget?
A warehouse manager has allowed 12 minutes of packing labor per order in next month’s staffing plan. A random sample of 25 orders averaged 11.2 minutes. One took 16 minutes. The finance director wants to cut the allowance to 11.2; the shift supervisor points to the 16-minute order and wants to raise it. What does each number tell the manager?
Both businesses and all figures in this lesson are fictional. Assume the warehouse will handle the same mix of orders next month. Its budget concerns average labor per order across the shift. It makes no promise about the time a particular customer’s order will take.
Recall Lesson 044: A delivery dashboard stopped its clock when the truck parked. The customer cared about when the parcel reached the door. Would measuring ten times as many parking times resolve that difference?
Two kinds of variation
The manager wants the mean, or average, packing time for all comparable orders. The point estimate is her single best figure from the sample: 11.2 minutes. She knows that average exactly for the 25 orders she timed. What she does not know exactly is the average for the much larger group those orders represent.
Individual orders vary: folding a shirt takes less time than wrapping six glasses. The sample’s standard deviation summarizes how far individual packing times spread around their mean. It is 2.5 minutes here. Now imagine taking fresh random samples of 25 orders. Each sample will contain a different mix of easy and awkward jobs, so its average will move too. The standard error describes the spread of those sample averages.
For independent orders, estimate the standard error of the mean by dividing the sample standard deviation by the square root of the order count. The square root of 25 is 5, since 5 × 5 = 25. Here the estimated standard error is 2.5 ÷ 5 = 0.5 minute. If the manager timed 100 independent orders and found the same spread, it would fall to 2.5 ÷ 10 = 0.25 minute. More observations can sharpen the staffing estimate even if packing any one box remains just as unpredictable.
Put a range around the average
A confidence interval puts a lower and an upper limit around an estimate to express uncertainty from sampling. Start at the sample mean and go out in both directions by a margin of error. That margin equals the standard error times a multiplier chosen for the confidence level and method.
Assume the 25 packing times were independently sampled, with a roughly bell-shaped distribution and no extreme outliers. For a 95% interval for a mean with 25 observations and an estimated standard deviation, the appropriate multiplier from the statistical t distribution is about 2.064. It allows for having to estimate the spread from these same orders. The multiplier is supplied; the important part is what you do with the result. NIST explains the interval for a mean.
- Margin of error: 2.064 × 0.5 = 1.032 minutes per order.
- Lower limit: 11.2 − 1.032 = 10.168 minutes per order.
- Upper limit: 11.2 + 1.032 = 12.232 minutes per order.
Report the estimated mean as 11.2 minutes per order, with a 95% confidence interval of about 10.17 to 12.23 minutes. The 95% describes the method: if you repeatedly drew samples this way from the same process, about 95% of the intervals would contain its true mean. You cannot tell whether this particular interval is one that misses.
The current 12-minute allowance falls inside the interval. Cutting it to 11.2 would leave no room for the possibility that the sample happened to include some unusually quick jobs. Keep the allowance for now and time more orders across shifts before cutting staffing. Even 12 minutes leaves some risk of underbudgeting. The 16-minute order is real, but it tells you little by itself about the shift average. It can also fall outside the interval without invalidating it: the interval estimates a mean, not the time for the next box.
Exercise: Should a shop commit to buying leads?
Allow eight minutes and use a calculator. A stationery shop can sign a six-month agreement to buy inquiries at $24 each. Each inquiry that becomes a paid order within 30 days earns $100 after all costs that change with the order, before the inquiry fee. Each inquiry can generate at most one order. The owner will sign only if the lower end of the 95% interval for the conversion rate covers the break-even rate.
The owner randomly checks 400 distinct inquiries from a much larger pool from the same source. Exactly 100 became paid orders within 30 days. Every inquiry had the full 30 days to convert. Assume independent outcomes, unchanged terms and comparable future inquiries.
- Calculate the conversion rate and the rate needed to break even. At the sample rate, what would the shop expect to earn per inquiry after the fee?
- Calculate an approximate 95% confidence interval. Write the sample rate as a decimal, p. Its estimated standard error is √[p × (1 − p) ÷ 400], where √ means take the square root. Multiply by 1.96 for the margin, then subtract and add that margin to p.
- Decide whether to sign under the owner’s rule. A salesperson says the interval tells you how much the shop will earn on the very next inquiry. What are the two possible dollar outcomes instead?
The 1.96 multiplier is used for this large-sample approximation for a proportion, meaning a share of the total. It works best when both outcomes are well represented; here there are 100 purchases and 300 nonpurchases. Penn State gives the interval for a proportion.
Try the decision before reading the answer.
Answer: Price the uncertainty
The conversion estimate is 100 ÷ 400 = 0.25, or 25%. It takes 24 paid orders per 100 inquiries to cover the fees: 24 × $100 = $2,400, the same as 100 × $24. At the sample rate the expected net gain is (0.25 × $100) − $24 = $1 per inquiry.
The estimated standard error is √[(0.25 × 0.75) ÷ 400] ≈ 0.02165. The margin is 1.96 × 0.02165 ≈ 0.0424, or 4.24 percentage points. Subtract and add that margin to 25%: the approximate interval is 20.76% to 29.24%.
At the lower limit, 0.2076 × $100 − $24 = −$3.24 per inquiry on average. At the upper limit the same arithmetic gives +$5.24. The 24% break-even rate lies inside the interval. Decline the six-month agreement under the stated rule. The evidence is consistent with both a profitable source and a losing one. It does not establish that buying leads loses money.
On the next inquiry the shop either earns $100 − $24 = $76 or loses the $24 fee. Neither outcome is inside the range from −$3.24 to +$5.24. That range concerns the average gain across comparable inquiries. Before signing, the owner could seek a lower fee or collect a larger representative sample. If the seller plans to change where leads come from, investigate that change first.
Recall answer: More parking timestamps would estimate truck arrival more precisely while still missing the walk to the door. A narrow confidence interval cannot repair a clock that measures the wrong event.
Add to your evidence audit
Record the shop’s estimate, interval and decision threshold together: 25% converted, the approximate 95% interval was 20.76% to 29.24%, and the shop needs 24% to break even. Keeping the threshold beside the interval makes it clear why a positive estimate was insufficient to sign. Next you’ll distinguish evidence that a difference exists from evidence that it is large enough to matter.
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