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**Hypothesis Testing For Proportions**

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**P-Values of a Hypothesis Test**

If the null-hypothesis is true, the P-value (probability value) of a hypothesis test is the probability of obtaining a sample statistic with a value as extreme or more extreme than the one determined from the sample data (farther away).

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And then… The smaller the P-Value, the more evidence there is to reject the null hypothesis. A very small P-Value indicates a rare occurrence. If the P-Value ≤ α, then you will reject the null hypothesis. If the P-Value > α, then you will fail to reject the null hypothesis.

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**Tests… If Ha contains <, the test is a left-tailed test.**

P is the area to the left of the test statistic. If Ha contains >, the test is a right-tailed test. P is the area to the right of the test statistic. If Ha contains ≠, the test is a two-tailed test. P is the area to the left of the negative test statistic, and P is the area to the right of the positive test statistic.

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Definitions Hypothesis tests for proportions occur (for example) when a politician wants to know the proportion of his or her constituents who favor a certain bill or when a quality assurance engineer tests the proportion of parts which are defective. Z-Test for a Proportion P: A statistical test for a population proportion P. It can be used when np ≥ 5 and nq ≥ 5. (q is 1-p) A test statistic is the sample proportion p-hat. The standardized test statistic is z…formula to follow later.

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**Steps to follow for a test…**

Verify that np ≥ 5 and nq ≥ 5. If these are true, the distribution for p-hat will be normal and you can continue; otherwise you cannot use normal distribution for the problem. State the claim…Identify null and alternative hypotheses. Specify the level of significance (α). Sketch the sampling distribution (make a curve).

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More steps… Determine any critical values (see next slide). These will be borders between rejection regions and non-rejection regions (below). They will be the same values each time. Determine any rejection regions. These are a range of values for which the Ho is not probable. If a test statistic falls into this region, Ho is rejected. A critical value separates the rejection region from the non-rejection region.

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**Critical Values Tailed Significance Level Critical Value Left 0.10**

-1.28 Right 1.28 Two ±1.645 0.05 -1.645 1.645 ±1.96 0.01 -2.33 2.33 ±2.575

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**More steps… Find the z-score (standard score): Make a decision to**

reject or fail to reject Ho. Interpret the decision in the context of the original claim.

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Example… A medical researcher claims that less than 20% of adults in the U.S. are allergic to a medication. In a random sample of 100 adults, 15% say they have such an allergy. At α = 0.01, is there enough evidence to support the researcher’s claim? n = 100, p = 0.20, q = 0.80 1. np = 20, nq = 80…you can continue. 2. Ho: p ≥ 0.2, Ha: p < 0.2

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Example (cont’d)… Since Ha is <, this is a left-tailed test, and since α = 0.01, we will be using the critical value as (they use the symbol zo for this). See drawing on board for sketch. The rejection region is z < The standardized test statistic (z) is:

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Example (cont’d)… Since z = -1.25, and this is not in the rejection region, you should decide not to reject the null hypothesis. Interpretation: There is not enough evidence to support the claim that less than 20% of adults in the U.S. are allergic to the medication.

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In your groups… USA Today reports that 5% of US adults have seen an extraterrestrial being. You decide to test this claim and ask a random sample of 250 U.S. adults whether they have ever seen an extraterrestrial being. Of those surveyed 8% reply yes. At α = 0.01, is there enough evidence to reject the claim? Your group is to complete and document all the steps to come to the final answer. This will be turned in.

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**There will be some on Friday **

Homework… There will be some on Friday

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