In inferential statistics, we make inference about the population using the sample statistic. The field of statistical inference has two subdivisions: estimation and hypothesis testing. Estimation helps in finding the confidence interval around a point estimate. It tells us about the value of the parameter. A hypothesis test helps in comparing the value of the parameter to some value. It is defined as a statement about one or more populations.
The process of hypothesis testing involves the following steps:
(1) State the hypotheses: We have two hypotheses - Null hypothesis and Alternative hypothesis. The null hypothesis (H0) is the hypothesis to be tested. It is a proposition that is considered true unless the hypothesis test gives us convincing evidence to reject it. If the null hypothesis is rejected, then the alternative hypothesis (Ha) is accepted. The null hypothesis will always have an "equal to" sign with it. It could be one of greater than or equal to, equal to, and less than or equal to some hypothesized value. The alternative hypothesis won't contain the "equal to" sign. It will be the exact opposite of the null hypothesis.
For example, if you want to check whether the returns earned by a mutual fund is greater than or equal to 10 percent, equal to 10 percent, or less than or equal to 10 percent then following would be our hypothesis for each case.
Greater than or equal to 10 percent: H0: R≥10 percent; Ha: R<10 percent
Equal to 10 percent: H0: R=10 percent; Ha: R≠ 10 percent
Less than or equal to 10 percent: H0: R≤ 10 percent; Ha: R>10 percent
When we have an equal to sign in the null hypothesis, then the hypothesis test is called a two-tailed test. When we have a greater than or equal to sign or less than or equal to sign in the null hypothesis, then the hypothesis test is called a one-tailed test.
It is easier to understand the tailed test by looking at the alternative hypothesis. When the alternative hypothesis has a sign "not equal to," then that is a two-tailed hypothesis because the null hypothesis could be either in the left tail or the right tail.
When the alternative hypothesis has a sign <, then the null hypothesis would be rejected in the left tail. It is also called as a left tailed test.
When the alternative hypothesis has a sign >, then the null hypothesis would be rejected in the right tail. It is also called as a right-tailed test.
(2) Identify the appropriate test statistic and its probability distribution: The second step in the hypothesis testing involves the identification of the appropriate test statistic. For checking the mean of a normally distributed population, the statistic could either be z-statistic (if the population variance is known) or t-statistic (if the population variance is unknown). For comparing the variances, we can use chi-square test statistic or F-statistic.
The calculated value of the test statistic is used in determining whether the null hypothesis would be rejected or not. We don't say that the null hypothesis is accepted. Either we say that the null hypothesis is rejected or we fail to reject the null hypothesis.
Test Statistic = (Sample statistic - Value of population parameter under null hypothesis)/Standard error of the sample statistic
(3) Specify the significance level: Whether a null hypothesis is rejected or could not be rejected depends on the significance level of the hypothesis test. The calculated test statistic is compared with the critical value of test statistic. The critical value of test statistic is based on the significance level.
(4) State the decision rule: The decision rule depends on the whether the test is a one-tailed test or two-tailed test. For a one-tailed test, all the extreme values lie in one tail only. For a two-tailed test, half of the extreme values lie in the right tail, and the other half extreme values lie in the left tail.
For two-tailed test: Reject the null hypothesis if the test statistic is not between the positive and negative critical values. For a 5 percent level of significance, we fail to reject the null hypothesis if t0.025 > test statistic > -t0.025. Reject the null hypothesis if test statistic >t0.025 or test statistic <-t0.025.
For one-tailed test: Reject the null hypothesis if the test statistic is greater than the positive critical value for a right-tailed test. Reject the null hypothesis if the test statistic is lower than the negative critical value for a left tailed test. For a 5 percent level of significance, reject the null hypothesis if test statistic >t0.05 (for right-tailed test) or test statistic < -t0.05 (for left-tailed test)
(5) Collect the data and calculate the test statistic: The next step involves the collection of sample data and calculating the sample statistic. The test statistic is calculated using the sample statistic.
(6) Make the statistical decision: The test statistic calculated in the previous step is compared to the critical value of test statistic depending on the level of significance, and a decision is based as per the decision rule discussed in step 4.
(7) Make the economic or investment decision: The last step of the hypothesis testing involves making the economic or investment decision based on the statistical result. The statistically significant result might not be economically significant due to the presence of factors like taxes, transaction costs, and risk.
Next LOS: Type I and Type II errors, power of a test, decision rule, relationship between confidence interval and hypothesis tests
