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**The Language of Hypothesis Testing Section 9.1**

Alan Craig

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Objectives 9.1 Determine the null and alternative hypotheses from a claim Understand Type I and Type II errors Understand the probability of making Type I and Type II errors State conclusions to hypothesis tests

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Hypothesis A hypothesis is a statement or claim regarding a characteristic of one or more populations.

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Examples The majority of Americans believe that illegal immigrants currently residing in the United States should be allowed to remain and have a path to citizenship. Lipitor can significantly reduce the risk of cardiovascular events compared with Simvastatin. 13% of Americans would be bothered if they had a hotel room on the 13th floor. The life of a GE Appliance A15 light bulb is 1500 hours. Tide has 3 times the stain removal power. A 20 fluid ounce bottle of Coca-Cola contains 20 fl. oz. A Toyota Prius gets 60 mpg in city driving. Most students love math.

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Hypothesis Testing Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of one or more populations.

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**Steps in Hypothesis Testing**

1 3 2 A claim is made Evidence (sample data) is collected to test the claim Data are analyzed to support or refute the claim

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**Two Types of Hypotheses**

The null hypothesis, denoted H0 (H-naught), is a statement to be tested. The null hypothesis is assumed true until evidence indicates otherwise. In this chapter, it will be a statement regarding the value of a population parameter. The alternative hypothesis, denoted H1 (H-one), is a claim to be tested. We are trying to find evidence for the alternative hypothesis. In this chapter, it will be a claim regarding the value of a population parameter.

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**Stating Hypotheses We consider three ways of stating hypotheses.**

Two-tailed test (equal versus not equal) H0: parameter = some value H1: parameter ≠ some value Left-tailed test (equal versus less than) H1: parameter < some value Right-tailed test (equal versus greater than) H1: parameter > some value

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Stating Hypotheses Note: Left-tailed and right-tailed tests are called one-tailed tests. The type of hypothesis statement depends on the researcher’s concerns. A 20 fluid ounce bottle of Coca-Cola contains 20 fl. oz. A quality control engineer would state H1: m ≠ 20. (two-tailed test) A consumer advocate would state H1: m < 20. (one-tailed test)

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**Type I and Type II Errors**

Type I: We reject H0, but H0 is true . Type II: We do not reject H0, but H1 is true. Reality H0 is True H1 is True Do Not Reject H0 Conclusion Correct Conclusion Type II Error Type I Error Correct Conclusion

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**Type I and Type II Errors: Trial Analogy**

H0: the defendant is innocent . H1: the defendant is guilty. Reality Innocent Guilty Innocent Verdict Guilty Correct Verdict Type II Error Type I Error Correct Verdict

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**Type I and Type II Errors**

William Blackstone writing on English law in 1769:”It is better that ten guilty persons escape, than that one innocent suffer.” So which of the two types of error did Blackstone prefer for criminal cases? The State of Georgia will pay $1.2 million for a Type I error in which a man spent 24 years in prison for a crime he did not commit.

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Level of Significance The level of significance, a, is the probability of making a Type I error. = P(Type I error) = P(rejecting H0 when H0 is true) = P(Type II error) = P(not rejecting H0 when H1 is true) a is chosen BEFORE data are collected.

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Level of Significance We choose the level of significance, a, based on the severity of the consequences of making a Type I error. Typical levels of significance are a=0.10, 0.05, and 0.01.

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**Level of Significance a and b are inversely related.**

If we decrease a = P(rejecting H0 when H0 is true), then we are increasing b = P(not rejecting H0 when H1 is true). In a criminal trial, we want a guilty verdict only when jurors are convinced the defendant is guilty “beyond a reasonable doubt.”

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**Writing the Conclusion**

We never accept the null hypothesis. We fail to reject it. The sample evidence cannot prove the null hypothesis is true. It can only show that it could be true. For example, verdicts in criminal trials are “not guilty” rather than “innocent.”

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**Writing the Conclusion**

If we reject H0, then we conclude that there is sufficient evidence to support the claim made in H1. If we do not reject H0, we conclude that there is not sufficient evidence to support the claim made in H1.

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Example: #10, 18, p. 390 According to the USDA, 78.2% of females between 12 and 19 years old get the USDA’s recommended daily allowance of protein. A nutritionist believes that the percentage at her boarding school is more than 78.2% What are the null and alternative hypotheses? What would it mean to make a Type I error? What would it mean to make a Type II error? State the conclusion if the null hypothesis is rejected.

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**Example: #10, 18, p. 390 What are the null and alternative hypotheses?**

According to the USDA, 78.2% of females between 12 and 19 years old get the USDA’s recommended daily allowance of protein. A nutritionist believes that the percentage at her boarding school is more than 78.2% What are the null and alternative hypotheses? H0: p = 0.782, H1: p > 0.782 Is this a one-tailed or two-tailed test?

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**Example: #10, 18, p. 390 What would it mean to make a Type I error?**

According to the USDA, 78.2% of females between 12 and 19 years old get the USDA’s recommended daily allowance of protein. A nutritionist believes that the percentage at her boarding school is more than 78.2% What would it mean to make a Type I error? The sample evidence leads us to believe that p > when it is not. What would it mean to make a Type II error? The sample evidence leads us to believe that p = when p is actually greater than

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Example: #10, 18, p. 390 According to the USDA, 78.2% of females between 12 and 19 years old get the USDA’s recommended daily allowance of protein. A nutritionist believes that the percentage at her boarding school is more than 78.2% State the conclusion if the null hypothesis is rejected. There is sufficient evidence to support the claim that females between 12 and 19 years old at the nutritionist’s boarding receive more than the recommended daily allowance of protein.

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