3.3 Introducing Statistics: Why Be Normal?

Why sampling distributions matter

When conditions are met, the sampling distribution of a sample proportion (p̂) is approximately Normal. This lets us use the Normal model to calculate probabilities, build confidence intervals, and run significance tests. Without approximate Normality, our probability calculations based on z-scores would be invalid.

Checking conditions for Normality

Use the Large Counts Condition: both np ≥ 10 and n(1-p) ≥ 10 (or using p̂ if p is unknown). This ensures the sampling distribution of p̂ is roughly symmetric and bell-shaped rather than skewed, so Normal-based methods (z-scores, z-intervals, z-tests) give accurate results.

What happens if conditions fail

If np or n(1-p) is less than 10, the sampling distribution may be skewed, especially when p is close to 0 or 1. In that case, Normal approximations are unreliable — probabilities and interval estimates calculated using z-scores could be inaccurate, so alternative methods or larger samples are needed.

Key terms

Large Counts Condition

np ≥ 10 and n(1-p) ≥ 10; ensures the sampling distribution of p̂ is approximately Normal.

Why check Normality conditions?

To justify using z-scores and the Normal model for accurate probability calculations, confidence intervals, and significance tests.

Effect of p near 0 or 1 with small n

The sampling distribution of p̂ becomes skewed rather than approximately Normal.

Sampling distribution of p̂

The distribution of sample proportions from repeated samples of the same size; approximately Normal when conditions are met.

Consequence of failing Large Counts Condition

Normal-based probability and interval calculations may be inaccurate due to skewness.

Try an AP-style question

Which condition checks whether the sampling distribution of p̂ is approximately Normal?

1. p is exactly 0.5 in the population
2. The population itself is Normal
3. np ≥ 10 and n(1 − p) ≥ 10
4. n ≥ 30 for every sample