Two-tailed tests

In a nutshell

The procedure of conducting two-tailed hypothesis tests is very similar to that of one-tailed tests. The main difference is that the relevant probability is compared to half of the significance level.

How to conduct a two-tailed test

To conduct a two-tailed test, follow this procedure.

procedure

Deciding the relevant probability

Because the test is two-tailed, the alternative hypothesis does not indicate the inequality for the relevant probability. Instead, use the mean:

The mean of a binomial distribution with parameters $$n$$ and $$p$$ is $$np$$.

If the observed value is less than the mean, then the relevant probability will have a $$\leq$$ sign.​

Example 1

A casino is suspected of using weighted dice. A particular die is tested for whether or not it is biased towards or against landing on $$1$$. The die is rolled $$30$$ times and it lands on $$1$$​ only once. Test, at the $$10\%$$​ significance level, whether or not the die is biased.

Define the test statistic and write down the hypotheses:

Let $$X$$ be the number of times the die lands on $$1$$.
​$$X\sim B(30,p)$$​​

​$$H_0:p=\dfrac{1}{6}$$​​

​$$H_1:p\neq\dfrac{1}{6}$$​​

Assume the null hypothesis to be true:

​$$p=\dfrac{1}{6}\Rightarrow X\sim B(30,\dfrac16)$$​​

Calculate the relevant probability:

The mean of this binomial distribution is $$np=30\times\dfrac16=5$$​. Because $$1<5$$, the relevant probability is $$P(X\leq 1)$$.

​$$P(X\le1)=0.0295$$​​

Compare this with half of the significance level:

​$$0.0295 < 0.1\div2=0.05$$​, so reject the null hypothesis.

Conclude based on the context of the question:

There is sufficient evidence to suggest that the die is biased at the 10% level of significance.

Note: It is also possible to find the critical values of the test and see whether or not the observed value lies in the critical region, but the procedure above is quicker.

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