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

1.	Define the test statistic and model it with the appropriate binomial distribution.
2.	Write down the null and alternative hypotheses.
3.	Assume the null hypothesis to be true.
4.	Calculate the relevant cumulative probability and compare it with the significance level. If the probability is less than the significance level, reject the null hypothesis. If the probability is greater than the significance level, do not reject the null hypothesis.
5.	Write a conclusion based on the context of the question.

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.