SECTION 7.1 Introduction to Hypothesis Testing 349 Stating a Hypothesis A statement about a population parameter is called a statistical hypothesis. To test a population parameter, you should carefully state a pair of hypotheses— one that represents the claim and the other, its complement. When one of these hypotheses is false, the other must be true. Either hypothesis—the null hypothesis or the alternative hypothesis—may represent the original claim. 1. A null hypothesis H0 is a statistical hypothesis that contains a statement of equality, such as …, =, or Ú. 2. The alternative hypothesis Ha is the complement of the null hypothesis. It is a statement that must be true if H0 is false and it contains a statement of strict inequality, such as 7, ≠, or 6. The symbol H0 is read as “H sub-zero” or “H naught” and Ha is read as “H sub-a.” DEFINITION To write the null and alternative hypotheses, translate the claim made about the population parameter from a verbal statement to a mathematical statement. Then, write its complement. For instance, if the claim value is k and the population parameter is m, then some possible pairs of null and alternative hypotheses are eH0: m … k Ha: m 7 k , e H0: m Ú k Ha: m 6 k , and e H0: m = k Ha: m ≠k . Regardless of which of the three pairs of hypotheses you use, you always assume m = k and examine the sampling distribution on the basis of this assumption. Within this sampling distribution, you will determine whether or not a sample statistic is unusual. The table shows the relationship between possible verbal statements about the parameter m and the corresponding null and alternative hypotheses. Similar statements can be made to test other population parameters, such as p, s, or s 2. Verbal Statement H0 The mean is . . . Mathematical Statements Verbal Statement Ha The mean is . . . . . . greater than or equal to k. . . . at least k. . . . not less than k. . . . not shorter than k. eH0: m Ú k Ha: m 6 k . . . less than k. . . . below k. . . . fewer than k. . . . shorter than k. . . . less than or equal to k. . . . at most k. . . . not more than k. . . . not longer than k. eH0: m … k Ha: m 7 k . . . greater than k. . . . above k. . . . more than k. . . . longer than k. . . . equal to k. . . . k. . . . exactly k. . . . the same as k. . . . not changed from k. eH0: m = k Ha: m ≠k . . . not equal to k. . . . different from k. . . . not k. . . . different from k. . . . changed from k. Picturing the World A study was done on the effect of a drug for the treatment of obesity due to genetic variants. The study used a random sample of 35 patients with severe obesity. At the end of the study, the patients had a mean reduction in baseline body weight of 3.7%. So, it is claimed that the mean reduction in baseline body weight is 3.7% for all patients with severe obesity who take the drug. (Adapted from Rhythm Pharmaceuticals, Inc.) Determine a null hypothesis and alternative hypothesis for this claim. Study Tip The term null hypothesis was introduced by Ronald Fisher (see page 35). If the statement in the null hypothesis is not true, then the alternative hypothesis must be true.
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