A statistical hypothesis is an assumption about a population parameter. For example, we may assume that the mean height of a male in the U.S. is 70 inches. The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter.
A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.
The Two Types of Statistical Hypotheses
To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.
There are two types of statistical hypotheses:
The null hypothesis, denoted as H0, is the hypothesis that the sample data occurs purely from chance.
The alternative hypothesis, denoted as H1 or Ha, is the hypothesis that the sample data is influenced by some non-random cause.
A hypothesis test consists of five steps:
1. State the hypotheses.
State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.
2. Determine a significance level to use for the hypothesis.
Decide on a significance level. Common choices are .01, .05, and .1.
3. Find the test statistic.
Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)
4. Reject or fail to reject the null hypothesis.
Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.
The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.
5. Interpret the results.
Interpret the results of the hypothesis test in the context of the question being asked.
The Two Types of Decision Errors
There are two types of decision errors that one can make when doing a hypothesis test:
Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha, and denoted as α.
Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta, denoted as β.
One-Tailed and Two-Tailed Tests
A statistical hypothesis can be one-tailed or two-tailed.
A one-tailed hypothesis involves making a “greater than” or “less than ” statement. For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.
A two-tailed hypothesis involves making an “equal to” or “not equal to” statement. For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.
Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.