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Unformatted text preview: Lecture 23 Nancy Pfenning Stats 1000 Chapter 11: Testing Hypotheses About Proportions Recall: last time we presented the following examples: 1. In a group of 371 Pitt students, 42 were left-handed. Is this significantly lower than the proportion of all Americans who are left-handed, which is .12? 2. In a group of 371 students, 45 chose the number seven when picking a number between one and twenty at random. Does this provide convincing statistical evidence of bias in favor of the number seven, in that the proportion of students picking seven is significantly higher than 1 / 20 = . 05? 3. A university has found over the years that out of all the students who are offered admission, the proportion who accept is .70. After a new director of admissions is hired, the university wants to check if the proportion of students accepting has changed significantly. Suppose they offer admission to 1200 students and 888 accept. Is this evidence of a change from the status quo? Each example mentions a possible value for p , which would indicate no difference/no change/status quo. The null hypothesis H states that p equals this traditional value. In contrast to the null hypothesis, each example suggests that an alternative may be true: a significance test problem always pits an alternative hypothesis H a against H . H a proposes that the proportion differs from the traditional value p H a rocks the boat/upsets the apple cart/ marches to a different drummer. A key difference among our three examples is the direction in which H a refutes H . In the first, it is suggested that the proportion of all Pitt students who are left-handed is less than the proportion for adults in the U.S., which is .12. In the second, we wonder if the proportion of students picking the number seven is significantly more than .05. In the third, we inquire about a difference in either direction from the stated proportion of .70. We can list our null and alternative hypotheses as follows: 1. H : p = . 12 H a : p < . 12 2. H : p = . 05 H a : p > . 05 3. H : p = . 70 H a : p = . 70 In general, we have H : p = p vs. H a : p < > = p Note that your textbook may have expressed the first two null hypotheses as H : p . 12 and H : p . 05. These expressions serve well as logical opposites to the alternative hypotheses, but our strategy to carry out a test will be to assume H is true, which means we must commit to a single value p at which to center the hypothesized distribution of p . Thus, we will write H : p = p in these notes. Alternatives with < or > signs are called one-sided alternatives ; with = they are two-sided . When in doubt, a two-sided alternative should be used, because it is more general....
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- Fall '06