One-tailed tests

In a nutshell

Conducting one-tailed hypothesis tests involves modelling the situation and test statistic, writing down the hypotheses and comparing the probabilities with the significance level.

How to conduct a one-tailed test

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

procedure

Example 1

A casino is suspected of using weighted dice. A particular die is tested for whether or not it is biased towards landing on $$1$$. The die is rolled $$30$$ times and it lands on $$1$$ a total of $$7$$ times. Is there sufficient evidence, at the $$5\%$$ significance level, that the die is biased towards landing on $$1$$​?

Define the test statistic and write down the hypotheses:

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

​$$H_0:p=\dfrac{1}{6}$$​​

​$$H_1:p>\dfrac{1}{6}$$​​

Assume the null hypothesis to be true:

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

Calculate the relevant probability:

The alternative hypothesis has a $$>$$ sign. So, the relevant cumulative probability is $$P(X\ge7)$$​.

​$$\begin{aligned}P(X\ge7)&=1-P(X\le6)\\&=1-0.7765\\&=0.2235\end{aligned}$$​​

Compare this with the significance level:

​$$0.2235 > 0.05$$​, so do not reject the null hypothesis.

Conclude based on the context of the question:

There is not enough evidence to suggest that the die is biased at the 5% level of significance.

Note: It is also possible to find the critical value 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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