SECTION 7.1 Introduction to Hypothesis Testing 351 Types of Errors and Level of Significance No matter which hypothesis represents the claim, you always begin a hypothesis test by assuming that the equality condition in the null hypothesis is true. So, when you perform a hypothesis test, you make one of two decisions: 1. reject the null hypothesis or 2. fail to reject the null hypothesis. Because your decision is based on a sample rather than the entire population, there is always the possibility you will make the wrong decision. For instance, you claim that a coin is not fair. To test your claim, you toss the coin 100 times and get 49 heads and 51 tails. You would probably agree that you do not have enough evidence to support your claim. Even so, it is possible that the coin is actually not fair and you had an unusual sample. But then you toss the coin 100 times and get 21 heads and 79 tails. It would be a rare occurrence to get only 21 heads out of 100 tosses with a fair coin. So, you probably have enough evidence to support your claim that the coin is not fair. However, you cannot be 100% sure. It is possible that the coin is fair and you had an unusual sample. Letting p represent the proportion of heads, the claim that “the coin is not fair” can be written as the mathematical statement p ≠ 0.5. Its complement, “the coin is fair,” is written as p = 0.5, as shown in the figure. p 0.52 0.51 0.48 0.49 0.50 Ha H0 Ha So, the null hypothesis is H0: p = 0.5 and the alternative hypothesis is Ha: p ≠0.5. (Claim) Remember, the only way to be absolutely certain of whether H0 is true or false is to test the entire population. Because your decision—to reject H0 or to fail to reject H0—is based on a sample, you must accept the fact that your decision might be incorrect. You might reject a null hypothesis when it is actually true. Or, you might fail to reject a null hypothesis when it is actually false. These types of errors are summarized in the next definition. A type I error occurs if the null hypothesis is rejected when it is true. A type II error occurs if the null hypothesis is not rejected when it is false. DEFINITION The table shows the four possible outcomes of a hypothesis test. Truth of H0 Decision H0 is true. H0 is false. Do not reject H0. Correct decision Type II error Reject H0. Type I error Correct decision
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