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University of Illinois, Chicago *

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361

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Statistics

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Feb 20, 2024

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Stat 361: Section 10.2; Section 10.3 Page 1 Section 10.2- Testing a Statistical Hypothesis Section 10.3 - The Use of p -value for Decision Making in Testing Hypotheses INTRODUCTION Q 1. What will we be doing in these sections? Goals: • Part of Step 3 of Hypothesis Testing. • Learn how to make decisions and draw conclusions as a result of the hypot- hesis test. • Investigate potential problems with performing a hypothesis test. Definition 1 (Test Statistic) . blank In Step 2 of Hypothesis Testing, we calculate something called a test statistic . A test statistic is a value calculated from data. We can either work with a test statistic directly from the data, or centered and re-scaled to enable comparisons. These test statistics enable us to determine whether to Reject H 0 or not. Q 2. How can we make decisions? CRITICAL REGION APPROACH Definition 2 (Critical Region (CR)) . blank A region where we would reject the null hypothesis. We have either 1 region or 2 regions (depending on our alternative hypothesis), where we would Reject H 0 . If we see a test sta- tistic value within this “rejection” region, we would make the decision to Reject H 0 . Indicated by: “Reject H 0 if ”. Definition 3 (Critical Value) . blank The value(s) where the “rejection” region(s) starts.
Stat 361: Section 10.2; Section 10.3 Page 2 Example 1 . A cold vaccine is known to be only 25% e ↵ ective after 2 years. To determine if a new vaccine is superior in providing protection against the same virus for a longer period of time, suppose 20 people are chosen at random and inoculated. Doctors believe that if more than 8 of those with the new vaccine surpass the 2-year period without contracting the virus, the new vaccine will be considered superior. (a) What are the hypotheses for this scenario? (b) Where does the 8 come from in the statement “Doctors believe...”? Solution: • It is somewhat arbitrary. • It appears reasonable as a modest gain in the number of people who have not gotten sick. • How many people would we expect to be protected? • 6 people and 7 people are still “reasonably” close to 5, so we choose a number of people slightly larger than this. • If you still felt that 8 were too close to 5, you could choose 9 instead. (c) When would we make the decision to Reject H 0 or Do Not Reject H 0 ?
Stat 361: Section 10.2; Section 10.3 Page 3 (d) How can we represent our critical region as a picture? How can we formally write down the critical region? Remark 1 . blank 1) For a 1-sided test, the sign on the critical region is the same as the sign on the alternative hypothesis. 2) For a 2-sided test, there are two critical regions.
Stat 361: Section 10.2; Section 10.3 Page 4 Example 2 . We are interested in the average weight of male college students at a certain college. We believe that they weight 68 kg ( ⇡ 150 pounds). We want to see if the average weight is di ↵ erent from this. (a) What are the hypotheses for this scenario? (b) What are some potential critical values? • If we do not want to Reject H 0 , then the sample mean should be “close” to our hypothesized value of 68. • If we Reject H 0 , then the sample mean should be “far” from our hypothesized value of 68. • Our critical values could potentially be 67 and 69. (c) How can we represent the critical region with a picture? When would we make the decision to Reject H 0 or Do Not Reject H 0 ?
Stat 361: Section 10.2; Section 10.3 Page 5 HOW TO FIND CRITICAL VALUES Q 3. How can we find critical values? • We have seen that critical values can be somewhat arbitrary. • Are there any guidelines as to how to determine which ones we should use? Q 4. What are the critical values and critical region for a 2-sided (tailed) test? Hypothesis Test Setup: H 0 : p = p 0 versus H 1 : p 6 = p 0 Preset ↵ : Suppose ↵ = 0 . 10. Test Statistic Values (establish a critical region based on ↵ ):
Stat 361: Section 10.2; Section 10.3 Page 6 Q 5. What are some other potential critical values for a 2-sided test involving Z as a test statistic? ↵ ↵ / 2 Critical Values 0.09 0.045 ± z 0 . 045 = ± invnorm (1 - 0 . 045) = ± 1 . 70 0.08 0.04 ± z 0 . 040 = ± invnorm (1 - 0 . 040) = ± 1 . 75 0.07 0.035 ± z 0 . 035 = ± invnorm (1 - 0 . 035) = ± 1 . 81 0.06 0.03 ± z 0 . 030 = ± invnorm (1 - 0 . 030) = ± 1 . 88 0.05 0.025 ± z 0 . 025 = ± invnorm (1 - 0 . 025) = ± 1 . 96 Example 3 . Suppose we are testing H 0 : p = 0 . 4 versus H 1 : p 6 = 0 . 4 . The success / failure condition is met. If ↵ = 0 . 07, then the critical values are ± 1 . 81. The critical region is Reject H 0 if z < - 1 . 81 OR if z > 1 . 81 . (a) If we find, from our data, that z = 1 . 85, what decision should we make? (b) If we find, from our data, that z = - 1 . 75, what decision should we make?
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