Stating Hypothesis

A statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true.

There are two types of statistical hypotheses for each situation: the null hypothesis and the alternative hypothesis.

The null hypothesis, denoted by $H_0$, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters.

The type of null hypothesis we will be concerned with involve means. i.e. we will always have a null hypothesis as

\[H_0:\text{ }\mu=k\]

The alternative hypothesis, denoted $H_1$, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters.

The alternative hypothesis will come in 3 different flavors:

$H_1:$ $\mu\neq k$ Two-tailed test
$H_1:$ $\mu < k$ Left-tailed test
$H_1:$ $\mu>k$ Right-tailed test

Example(s):

1) A medical researcher is interested in finding out whether a new medication will have any undesirable side effects. The researcher is particularly concerned with the pulse rate of the patients who take the medication. Will the pulse rate increase, decrease, or remain unchanged after a patient takes the medication?

Since the researcher knows that the mean pulse rate for the population under study is 82 beats per minute, the hypotheses for this situation are

\[H_0:\text{ }\mu=82\text{ and }H_1:\text{ }\mu\neq 82\]

2) A contractor wishes to lower heating bills by using a special type of insulation in houses. If the average of the monthly heating bills is $78, her hypotheses about heating costs with the use of insulation are

\[H_0:\text{ }\mu=78\text{ and }H_1:\text{ }\mu<78\]

A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected.

The numerical value obtained from a statistical test is called the test statistic