10.2 Estimating Population Totals, Means & Percentages from Survey Data

10.2 Estimating Population Totals, Means & Percentages from Survey Data

Skill 4 of Competency 4 names three targets — population total, mean, and percentage — all estimated from a sample. This skill was added when Florida redeveloped test 025 and is absent from older prep material, so it deserves deliberate attention.

The core logic

A random sample resembles its population. Therefore whatever proportion you observe in the sample is your best estimate of the proportion in the population, and multiplying by the population size converts it into a count.

+---------------------------------------------------------------------------+
|   sample proportion  p-hat = (count in sample) / (sample size)            |
|                                                                           |
|   ESTIMATED POPULATION PERCENTAGE  =  p-hat x 100%                        |
|   ESTIMATED POPULATION TOTAL       =  p-hat x N                           |
|   ESTIMATED POPULATION MEAN        =  sample mean (used directly)         |
|   ESTIMATED POPULATION TOTAL       =  sample mean x N                     |
|      (for a quantity, e.g. total hours)                                   |
+---------------------------------------------------------------------------+

Estimating a percentage

In a random sample of 250 voters, 145 support a referendum. Estimate the percentage of all voters who support it. p̂ = 145/250 = 0.58, so the estimate is 58%.

Note what does not appear: the size of the voting population. A proportion is a ratio within the sample, so N is irrelevant until you convert to a count. Items include a population figure specifically to see whether you divide by the wrong number.

Estimating a population total

A school has 1,400 students. In a random sample of 175, exactly 63 walk to school. Estimate how many students in the school walk. p̂ = 63/175 = 0.36, so the estimate is 0.36 × 1,400 = 504 students.

The proportional set-up is equivalent and often clearer:

63/175 = x/1,400 → 175x = 88,200 → x = 504 ✓

Estimating a mean and a total quantity

The sample mean is the estimate of the population mean — no scaling.

A random sample of 40 households in a town of 5,200 households uses an average of 310 gallons of water per day. Estimated population mean: 310 gallons per household per day. Estimated total daily use: 310 × 5,200 = 1,612,000 gallons.

The distinction matters. Multiplying the mean by N gives a total; the mean itself needs no adjustment. Items offer 310 × 5,200 as the answer to a mean question and 310 as the answer to a total question, so read which is requested.

A sample of 60 books from a library's collection of 24,000 has a mean of 284 pages. Estimate the total number of pages in the collection. 284 × 24,000 = 6,816,000 pages.

Capture-recapture

A classic technique for estimating an unknown population size N, common in ecology and a frequent item context.

1. Capture, mark, and release M individuals.
2. Later capture a second sample of size n.
3. Count m, the marked individuals in the second sample.
4. Assume the marked proportion in the second sample matches the marked proportion in the population:

M/N = m/n, so N = (M × n)/m

Biologists tag 80 fish and release them. Later they catch 120 fish, of which 15 are tagged. N = (80 × 120)/15 = 9,600/15 = 640 fish.

The reasoning: tagged fish are 15/120 = 12.5% of the second catch, so the 80 tagged fish should be about 12.5% of the whole population, giving 80/0.125 = 640. ✓

The method assumes the marked individuals mixed back into the population and that no births, deaths, or migrations occurred between the samples — assumptions items sometimes ask you to identify.

Interpreting and limiting the estimate

Every estimate from a sample carries sampling variability: a different random sample would give a different number. Two consequences the exam tests:

1. Never report an estimate as an exact fact. "About 504 students walk to school" is correct; "exactly 504 students walk" is not.
2. Larger random samples give more stable estimates. Doubling the sample size does not double the accuracy, but it does reduce the spread of estimates across repeated samples.

And the constraint from section 10.1 still governs everything: all of this arithmetic is valid only if the sample was randomly selected from the target population. Scaling up a convenience sample produces a precisely computed but systematically wrong number.

A newspaper surveys 400 people leaving a farmers market and finds 72% buy organic produce weekly. It reports that "72% of city residents buy organic produce weekly." The arithmetic is fine; the inference is not. Farmers market shoppers are not representative of all residents, so the estimate is biased upward regardless of the sample size.

Comparing two estimates

Items sometimes give two samples and ask which supports a stronger conclusion.

Sample A: 40 students, 22 prefer the later start time (55%). Sample B: 400 students, 216 prefer the later start time (54%). Both estimates are close, but Sample B is more reliable because larger random samples vary less. The near-agreement between the two also increases confidence that the true value is near 54–55%.