Mathwords logoMathwords

Conditions for Inference — Definition, Formula & Examples

Conditions for inference are the checks you run on your data and how it was collected before you trust a confidence interval or significance test. If a condition fails, the formulas can give misleading answers.

Conditions for inference are the assumptions under which a procedure's sampling distribution model holds. The AP Statistics conditions are: (1) Random: data come from a random sample or a randomized experiment. (2) Independence: when sampling without replacement, n≤0.10Nn \le 0.10N (the 10% condition). (3) Normal/Large Counts: the sampling distribution is approximately normal. For a proportion this means np≥10np \ge 10 and n(1−p)≥10n(1-p) \ge 10. For a mean it means n≥30n \ge 30, or the population is approximately normal. Regression inference uses its own set of conditions: linearity, independence, normal residuals, equal spread, and random data.

Key Formula

n≤0.10Nnp^≥10n(1−p^)≥10n \le 0.10N \qquad n\hat{p} \ge 10 \qquad n(1-\hat{p}) \ge 10
Where:
  • nn = sample size
  • NN = population size
  • p^\hat{p} = sample proportion. For a significance test, use the hypothesized value p0p_0 instead.

Worked Example

Problem: A random sample of 200 students from a school of 3,000 finds that 60 students prefer a later start time. Check the conditions for a one-proportion z-interval.
Step 1: Random: the problem states the students were chosen by random sampling, so this condition is met. Independence (10% condition): the sample must be no more than 10% of the population.
0.10(3000)=300,200≤3000.10(3000) = 300, \quad 200 \le 300
Step 2: Large Counts: find the sample proportion, then check that the expected successes and failures are each at least 10.
p^=60200=0.3,np^=200(0.3)=60,n(1−p^)=200(0.7)=140\hat{p} = \frac{60}{200} = 0.3, \quad n\hat{p} = 200(0.3) = 60, \quad n(1-\hat{p}) = 200(0.7) = 140
Step 3: Both counts are at least 10, so the sampling distribution of p^\hat{p} is approximately normal.
60≥10,140≥1060 \ge 10, \quad 140 \ge 10
Answer: All three conditions are met: Random, 10% (200 ≤ 300), and Large Counts (60 and 140 are both at least 10). You can proceed with the z-interval.

Why It Matters

Every AP Statistics free-response inference question awards points for checking conditions with the actual numbers from the problem. In clinical trials and polling, analysts also have to justify that their sampling method and sample size support their conclusions. Otherwise reviewers reject the results.

Common Mistakes

Mistake: Using the rule n≥30n \ge 30 to check normality for a proportion.
Correction: The n≥30n \ge 30 guideline applies to means. For proportions, check that the success and failure counts are each at least 10. A sample of 50 with only 3 successes fails, even though 50 is greater than 30.

Related Terms