Test for one proportion
Section of Hypothesis tests, part of Statistical inference
Test for one proportion
| Type | Section |
| Taught in | Hypothesis tests |
| Part | Statistical inference |
Ten people taste two yoghurts, one with an old sweetener and one with a new. Eight prefer the new one. Model the number preferring the new yoghurt as X\sim\operatorname{Bin}(10,p), and test H_0:p=0.5 against H_1:p>0.5.
Definition 1 (Binomial test) For X\sim\operatorname{Bin}(n,p) and H_0:p=p_0, the test statistic is X, with null distribution \operatorname{Bin}(n,p_0). At level \alpha, against p>p_0 reject when X\ge c, with c the smallest integer such that \mathbb{P}_{H_0}(X\ge c)\le\alpha. Against p<p_0 reject when X\le c, with c the largest integer such that \mathbb{P}_{H_0}(X\le c)\le\alpha. Against p\neq p_0, use both tails at \alpha/2 each.
Under H_0, \mathbb{P}(X\ge8)=0.055 and \mathbb{P}(X\ge9)=0.011. At level 0.05 the critical region is \{9,10\}, and x=8 is not in it, so H_0 is not rejected. The p-value is 0.055, which leaves real doubt about H_0. That a p-value of 0.055 and one of 0.045 lead to opposite decisions is a good reason to report the p-value rather than only the decision.
For large n,
Z=\frac{X-np_0}{\sqrt{np_0(1-p_0)}}
is approximately standard normal under H_0, and the normal critical values apply. The null standard error uses p_0, not \widehat p: the test asks how variable X would be if H_0 were true.
Used in
See also
Null and alternative hypotheses · Critical region · p-value · Type I and type II errors · Neyman–Pearson approach · Power function · One-sample t test · Two-sample t test