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Null Distribution — Definition, Formula & Examples

The null distribution is the pattern of results you would expect to see across many repeated samples if the null hypothesis were exactly true. You compare your actual result to it to judge whether that result is surprising.

The null distribution is the sampling distribution of a test statistic, computed under the assumption that the null hypothesis H0H_0 holds. Its tail areas beyond the observed statistic give the p-value. Its quantiles give the critical values for a chosen significance level α\alpha.

Key Formula

p^  ∼H0  approximately N ⁣(p0, p0(1−p0)n)\hat{p} \;\overset{H_0}{\sim}\; \text{approximately } N\!\left(p_0,\ \sqrt{\frac{p_0(1-p_0)}{n}}\right)
Where:
  • p^\hat{p} = sample proportion (the statistic being tested)
  • p0p_0 = population proportion claimed by the null hypothesis
  • nn = sample size
  • N(μ,σ)N(\mu, \sigma) = normal distribution with mean μ\mu and standard deviation σ\sigma. The approximation is reasonable when np0≥10np_0 \ge 10 and n(1−p0)≥10n(1-p_0) \ge 10.

Worked Example

Problem: You flip a coin 100 times and get 60 heads. Test H0:p=0.5H_0: p = 0.5 against Ha:p>0.5H_a: p > 0.5 using the null distribution of p^\hat{p}.
Step 1: Build the null distribution. Assume p=0.5p = 0.5. The conditions hold because 100(0.5)=50≥10100(0.5) = 50 \ge 10. Center the distribution at p0p_0 and use p0p_0 to find the standard deviation.
μp^=0.5,σp^=0.5(0.5)100=0.05\mu_{\hat{p}} = 0.5, \qquad \sigma_{\hat{p}} = \sqrt{\frac{0.5(0.5)}{100}} = 0.05
Step 2: Locate the observed result p^=0.60\hat{p} = 0.60 on the null distribution by computing the test statistic.
z=0.60−0.50.05=2z = \frac{0.60 - 0.5}{0.05} = 2
Step 3: Find the area of the null distribution at or beyond the observed value. This area is the one-sided p-value.
P(Z≥2)≈0.0228P(Z \ge 2) \approx 0.0228
Answer: If the coin were fair, you would get 60 or more heads only about 2.28% of the time. At α=0.05\alpha = 0.05, you reject H0H_0.

Why It Matters

Every hypothesis test in AP Statistics relies on a null distribution. The z-test uses a normal curve, the t-test uses a t-distribution with n−1n-1 degrees of freedom, and the chi-square test uses a chi-square distribution. On the AP exam's free-response questions, you often have to read a p-value off a simulated null distribution, such as a dotplot of results generated by assuming H0H_0.

Common Mistakes

Mistake: Using the sample proportion p^\hat{p} to compute the standard deviation in a one-proportion z-test, for example writing 0.6(0.4)/100\sqrt{0.6(0.4)/100}.
Correction: The null distribution assumes H0H_0 is true, so its center and spread both come from p0p_0. Use p0(1−p0)/n\sqrt{p_0(1-p_0)/n}. Save p^\hat{p} for the standard error of a confidence interval.

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