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# The Hypothesis-Testing Procedure for AP Statistics (page 2)

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If you stated a significance level in the second step of the process, the conclusion can be based on a comparison of the P-value with α. If you didn't state a significance level, you can argue your conclusion based on the value of the P-value alone: if it is small, you have evidence against the null; if it is not small, you do not have evidence against the null. Many statisticians will argue that you are better off to argue directly from the P-value and not use a significance level. One reason for this is the arbitrariness of the P-value. That is, if α = 0.05, you would reject the null hypothesis for a P-value of 0.04999 but not for a P-value of 0.05001 when, in reality, there is no practical difference between them.

The conclusion can be (1) that we reject H0 (because of a sufficiently small P-value) or (2) that we do not reject H0 (because the P-value is too large). We do not accept the null: we either reject it or fail to reject it. If we reject H0, we can say that we accept HA or, preferably, that we have evidence in favor of HA.

example: Consider, one last time, Todd and his claim that he can throw a ball 50 yards. His average toss, based on 50 throws, was 47.5 yards, and we assumed the population standard deviation was the same as the sample standard deviation, 8 years. A test of the hypothesis that Todd can throw the ball 50 yards on average against that alternative that he can't throw that far might look something like the following (we will fill in many of the details, especially those in the third part of the process, in the following chapters):
• Let μ be the true average distance Todd can throw a football. H0: μ = 50 (or H0: μ ≤ 50, since the alternative is one-sided) HA: μ < 50
• Since we know σ, we will use a z-test. We assume the 50 throws is an SRS of all his throws and the central limit theorem tells us that the sampling distribution of is approximately normal. We will use a significance level of α = 0.05.
• In the previous section, we determined that the P-value for this situation (the probability of getting an average as far away from our expected value as we got) is 0.014.
• Since the P-value < α (0.014 < 0.05), we can reject H0. We have good evidence that the true mean distance Todd can throw a football is actually less than 50 yards (note that we aren't claiming anything about how far Todd can actually throw the ball on average, just that it's likely to be less than 50 yards).

Practice problems for these concepts can be found at:

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