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AP STATISTICS · INFERENCE

Significance Tests

Is the evidence strong enough to reject the status quo?

17 min read 3 practice questions
A significance test weighs evidence against a claim. Start with a null hypothesis H0H_0 (no effect, no difference) and an alternative HaH_a. Then ask: if H0H_0 were true, how surprising is our data?
z=p^−p0p0(1−p0)nz = \frac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1 - p_0)}{n}}}

Test statistic for one proportion

The p-value is the probability of getting a result at least as extreme as ours, assuming H0H_0 is true. If the p-value is less than the significance level α\alpha (often 0.050.05), we reject H0H_0.

Worked example · Is the coin fair?

A coin lands heads 6262 times in 100100 flips. Test H0:p=0.5H_0: p = 0.5 vs. Ha:p≠0.5H_a: p \neq 0.5 at α=0.05\alpha = 0.05.
  1. 1
    p^=0.62\hat{p} = 0.62; SE=0.5(0.5)100=0.05SE = \sqrt{\frac{0.5(0.5)}{100}} = 0.05.
  2. 2
    z=0.62−0.50.05=2.4z = \frac{0.62 - 0.5}{0.05} = 2.4.
  3. 3
    Two-sided p-value ≈2(0.0082)=0.0164\approx 2(0.0082) = 0.0164.
  4. 4
    Since 0.0164<0.050.0164 < 0.05, reject H0H_0. There is convincing evidence the coin is not fair.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

1
A test gives p-value =0.03= 0.03. At α=0.05\alpha = 0.05, you should…
2
p^=0.56\hat{p} = 0.56, p0=0.5p_0 = 0.5, n=100n = 100. Compute the z test statistic.
3
A new drug actually works, but the study fails to reject H₀. This is a…

Numeric answers accept decimals or fractions like 3/4.