Conditions for Inference: Random, Normal, Independent
Apply the three standard conditions for inference: Random (data from random sample/randomized experiment), Normal (sampling distribution is approximately normal), Independent (10% condition: n ≤ 10% of population). State and check these on every inference problem.
A lesson of AP Statistics, in Sampling Distributions, about 10 minutes.
Conditions for Inference
By the end of this lesson, you will be able to state and verify the three standard conditions required for all inference procedures: Random, Normal, and Independent.
Why Conditions Matter
Inference procedures (confidence intervals and tests) are derived from the sampling distribution. The conditions ensure that the sampling distribution has the right shape (approximately normal) so that probabilities from the z or t table are accurate.
Violating a condition means the interval or p-value may be misleading.
The Three Conditions
1. Random
What it requires: Data must come from a random sample (observational study) or random assignment (experiment).
Why it matters: Random selection ensures the sample is representative of the population. Without randomness, the statistic may be biased and we can't generalize.
How to verify: Look at the study design. Was there an SRS? Random assignment? A convenience sample violates this condition.
If violated: Cannot generalize results to the population; cannot claim the sample is representative.
2. Normal (Approximate Normality of the Sampling Distribution)
What it requires: The sampling distribution of the statistic must be approximately normal.
For proportions ($\hat{p}$): Large Counts condition — $np \geq 10$ AND $n(1-p) \geq 10$
- For a CI: use $\hat{p}$: $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$
- For a test: use $p_0$: $np_0 \geq 10$ and $n(1-p_0) \geq 10$
For means ($\bar{x}$):
- If the population is normally distributed: the sampling distribution is exactly normal for any n
- If n ≥ 30: the CLT guarantees approximate normality (for any population shape)
- If n < 30: check for strong skewness or outliers in the sample data
Why it matters: The z and t critical values and p-values are based on the assumption of normality. If violated, the interval or p-value is inaccurate.
3. Independent
What it requires: Observations must be independent of one another.
How to verify: