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A two-sided test

A two-sided test is used when the alternative is that the proportion differs from the value in the null hypothesis, without the direction being given in advance. You calculate the same Z as in the one-sided test, and look at the absolute value. The course rule of thumb at the 5% level is to reject when the absolute value is greater than the table value 1.96. That boundary comes from the normal table, where Φ of 1.96 is 0.975. The Z-value itself can be calculated directly.

Z = \frac{X - np_0}{\sqrt{np_0(1-p_0)}}

the same statistic

|Z| > 1.96

two-sided 5% rule, from the table

Symbols

$p_0$	the proportion in the null hypothesis
$X$	observed count

Example

$H_0: p = 0.25$ , $n = 400$ and $X = 120$ .

$Z = \dfrac{120 - 100}{\sqrt{400 \cdot 0.25 \cdot 0.75}} \approx 2.31$ .

Two-sided: look at the absolute value of Z. One-sided: look at the sign as well.

Practise hypothesis testing for free →

← A test of a mean

Part of Mathematics S2: Hypothesis testing.