The idea, step by step
Define a binary outcome
Choose a success rule, for example travelling by bus at least three days per week. The population proportion p is the fraction of the target population satisfying that rule. A count x from n valid observations gives .
The population parameter is fixed for the chosen population and time. The estimate changes between samples. State the success rule and group, then express the estimate as a decimal or percentage.
Use the correct denominator
Missing responses are not automatically failures. Dividing yes responses by respondents estimates a respondent proportion; dividing by all selected people answers a different question. Either way, non-response may threaten a claim about the target population.
A binomial model needs fixed n, two outcomes, independent trials and a common success probability. Sampling a large fraction of a finite population without replacement can make the independence approximation unsuitable.
A common trap
The observed proportion equals the exact population proportion.
Instead: It is an estimate subject to sampling variation.
See the thinking
Count to estimate
138 of 240 randomly selected students meet a travel criterion. Estimate the population proportion.
- 138/240 = 0.575.
- Use sample notation for an estimate, not a known exact population value.
Missing answers
200 adults are selected; 150 respond and 90 say yes. Estimate the respondent proportion and give a limitation.
- 90/150 = 0.60.
- The 50 missing answers are unknown; random initial selection does not remove non-response bias.
Check your understanding
Find the sample proportion for 45 successes among 75 observations.
Show answer
0.6.
45/75 = 0.6.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Find the sample proportion for 84 successes among 140.2 marks
- Find the success count for n = 250 and sample proportion 0.36.2 marks
- A report writes p = 0.42 because 84 of 200 sampled people said yes. Correct the notation.2 marks
- Of 180 selected people, 120 respond and 72 say yes. Give the respondent estimate.2 marks
- Why might 20 responses from one household not be independent observations of community preference?2 marks
- Why should a success rule not be changed after inspecting results?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find the sample proportion for 84 successes among 140.2 marks
84/140 = 0.6.
0.6. - Find the success count for n = 250 and sample proportion 0.36.2 marks
250 × 0.36 = 90.
90. - A report writes p = 0.42 because 84 of 200 sampled people said yes. Correct the notation.2 marks
Reserve p for the population parameter.
estimates p. - Of 180 selected people, 120 respond and 72 say yes. Give the respondent estimate.2 marks
72/120 = 0.6.
72/180 instead measures the fraction selected who both responded and said yes.
0.6. - Why might 20 responses from one household not be independent observations of community preference?2 marks
The count alone does not establish binomial assumptions.
Shared circumstances and influence can correlate their answers. - Why should a success rule not be changed after inspecting results?2 marks
Pre-specify measurement rules and transparently explain justified changes.
A selective redefinition can favour a desired conclusion.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 Mathematical Methods: sampling distributions and confidence intervals for means and proportions. Mean tasks distinguish known population spread from sufficiently large-sample estimated-spread normal intervals. Repeated-interval computational simulation, approximation conditions, design and interpretation are included.
Official curriculum source ↗