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Normality Condition — Definition, Formula & Examples

The normality condition is a check you run before a confidence interval or significance test. It confirms that the normal model, or a t model, is a reasonable fit for the sampling distribution of your statistic.

For inference procedures based on z or t distributions, the sampling distribution of the statistic must be approximately normal. In AP Statistics this is verified as follows. For a mean, the population is approximately normal, or n≥30n \ge 30 (Central Limit Theorem), or a graph of a small sample shows no strong skew or outliers. For a proportion, the Large Counts condition np^≥10n\hat{p} \ge 10 and n(1−p^)≥10n(1-\hat{p}) \ge 10 holds. For a regression slope, the residuals are approximately normally distributed.

Key Formula

np^≥10andn(1−p^)≥10n\hat{p} \ge 10 \quad \text{and} \quad n(1-\hat{p}) \ge 10
Where:
  • nn = sample size
  • p^\hat{p} = sample proportion of successes. In a significance test, use the hypothesized value p0p_0 instead.
  • np^n\hat{p} = expected number of successes
  • n(1−p^)n(1-\hat{p}) = expected number of failures

Worked Example

Problem: A random sample of 200 students finds that 30% walk to school. Is the normality condition met for building a confidence interval for the true proportion?
Step 1: Find the number of successes in the sample.
np^=200(0.30)=60n\hat{p} = 200(0.30) = 60
Step 2: Find the number of failures in the sample.
n(1−p^)=200(0.70)=140n(1-\hat{p}) = 200(0.70) = 140
Step 3: Compare both counts to 10. Both 60≥1060 \ge 10 and 140≥10140 \ge 10 hold.
60≥10,140≥1060 \ge 10, \quad 140 \ge 10
Answer: Yes. Both counts are at least 10, so the sampling distribution of p^\hat{p} is approximately normal and a one-proportion z-interval is appropriate.

Why It Matters

On the AP Statistics free-response section, you lose credit if you skip a condition check or only name it without numbers. Stating the check with your actual values, such as "60 and 140 are both at least 10," is part of a complete answer. The same idea carries into college regression courses, where you check residual plots before trusting p-values.

Common Mistakes

Mistake: Writing "n ≥ 30, so the data are normal."
Correction: A large sample does not make the data normal. The Central Limit Theorem makes the sampling distribution of xˉ\bar{x} approximately normal. Say that the sampling distribution is approximately normal.

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