Testing a Hypothesis
In statistics, the researcher formulates a claim about a population of interest as a hypothesis, which will then undergo a procedure for testing the plausibility of the claim. For example, a researcher may wish to assess if a composite dental material has a translucency that is acceptably close to a specified target level. The researcher formulates the hypothesis that the mean translucency of this composite material \(\mu\) is equal to the specified target level \(\mu_0\). On the other hand, the researcher may wish to know if the translucency exceeds this specified target translucency level, in which case the hypothesis is \(\mu>\mu_0\). The way the hypothesis is defined will lead to either a two-sided test for \(\mu=\mu_0\), or a one-sided test for \(\mu>\mu_0\).
In a hypothesis testing framework there are two hypotheses that form mutually exclusive and exhaustive cases for the value of the parameter of interest. These are the null hypothesis H0 and the alternative hypothesis Ha. For example, if H0 is \(\mu=\mu_0\), then Ha is \(\mu \neq \mu_0\). On the other hand, if H0 is \(\mu > \mu_0\), then Ha is \(\mu \leq \mu_0\). The hypothesis testing procedure is one through which the value of a test statistic will be used by the researcher as evidence against the null hypothesis. For this reason, the research claim must be formulated so that it becomes the alternative hypothesis.
| Null Hypothesis (H0) | Alternative Hypothesis (Ha) | Number of Sides |
|---|---|---|
| \(\mu = \mu_0\) | \(\mu \neq \mu_0\) | Two-sided |
| \(\mu < \mu_0\) | \(\mu \geq \mu_0\) | One-sided |
| \(\mu > \mu_0\) | \(\mu \leq \mu_0\) | One-sided |
Subjecting scientific claims to rigorous testing ensures a high standard for adopting new claims. When a researcher makes a claim about a population or a process, that claim is subjected to an empirical test so as to convince a skeptic, who requires strong evidence, to accept this new claim. The test therefore proceeds as if to convince a scientific skeptic to agree with the researcher’s claim. The scientific skeptic begins by nullifying or refuting the researcher’s hypothesis, thereby assuming that the null hypothesis holds. An experiment is designed through which data is collected and a test statistic is produced, which will be used for the purpose of persuading the skeptic to change their mind. The researcher will use the value of the test statistic against the skeptic’s null hypothesis. If the value of this test statistic is sufficiently implausible to the position of the skeptic, who is assuming the null hypothesis is true, they will decide to not reject the researcher’s claim-the alternative hypothesis.
A test statistic calculated from the sample data measures how far the data diverges from what we would expect if the null hypothesis were true. The value of the test statistic show that the data are not consistent with the null hypothesis.
The p-value is the probability, computed assuming that the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. A small p-value is evidence against the null hypothesis because it says that the observed result would be unlikely to occur if the null hypothesis were true. A Large p-value fails to give evidence against the null hypothesis.
| Null Hypothesis is True in Reality | Null Hypothesis is False in Reality | |
|---|---|---|
| Do not reject null hypothesis | Correct Desision | Type II error \((\beta)\) |
| Reject null hypothesis | Type I error (\(\alpha\)) | Correct Decision \((Power = 1 – \beta)\) |
Type I Error: Rejecting the null hypothesis when it is in fact true. The probability of making a type I error is the significance level \(\alpha\).
The significance level \(\alpha\) is set by the researcher as the rate at which they will tolerate a type I error, namely rejecting the null hypothesis when it is in fact true. Commonly, the significance level alpha is set at \(\alpha=0.05\). The researcher will reject the null hypothesis if the p-value is less than the significance level \(\alpha\), because probability of rejecting the null hypothesis is less than the accepted threshold \(\alpha\).
Type II Error: Not rejecting the null hypothesis when it is in fact false. The probability of making a type II error is \(\beta\).
The power of a test is the probability of accepting the alternative hypothesis when in reality the alternative hypothesis is true. In other words, the power is the probability of not making a type II error, so power=\(1-\beta\).