Q4. Alpha again: Would the decision change at different significance levels?

Background

Topic: Hypothesis Testing & Significance Levels

This question examines how the choice of significance level () affects the outcome of a hypothesis test. It asks you to consider whether the conclusion would change if $\alpha$ were set to 0.10 or 0.01 instead of 0.05.

Key Terms and Formulas:

• Significance Level (): The probability of rejecting the null hypothesis when it is actually true (Type I error).

• P-value: The probability, under the null hypothesis, of obtaining a result as extreme or more extreme than the observed result.

• Decision Rule: If , reject ; otherwise, fail to reject $H_0$.

Step-by-Step Guidance

1. Recall that the environmentalists did not find evidence to reject the null hypothesis at . This means the P-value was greater than 0.05.

2. Consider what would happen if were increased to 0.10. Would a P-value greater than 0.05 but less than 0.10 change the decision?

3. Now, consider what would happen if were decreased to 0.01. Would a P-value greater than 0.05 also be greater than 0.01?

4. Think about how the threshold for rejecting the null hypothesis changes as increases or decreases, and how this affects the likelihood of making a Type I error.

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Q12. Alzheimer's: Hypothesis Testing and Errors

Background

Topic: Hypothesis Testing, Type I & II Errors, and Test Power

This question explores the application of hypothesis testing to medical diagnosis, specifically for Alzheimer's disease. It asks you to define hypotheses, interpret error types, and understand the concept of test power.

Key Terms and Formulas:

• Null Hypothesis (): The default assumption (e.g., the person is healthy).

• Alternative Hypothesis (): The claim we seek evidence for (e.g., the person has Alzheimer's).

• Type I Error (): Rejecting when it is true (false positive).

• Type II Error (): Failing to reject when is true (false negative).

• Power of a Test: (probability of correctly rejecting when is true).

Step-by-Step Guidance

1. For part (a), clearly state the null and alternative hypotheses in the context of Alzheimer's testing.

2. For part (b), interpret what a Type I error means in this medical context (think about the consequences of a false positive).

3. For part (c), interpret what a Type II error means (think about the consequences of a false negative).

4. For part (d), compare the seriousness of Type I and Type II errors in this context and explain your reasoning.

5. For part (e), use the given false negative rate to calculate the power of the test using the formula .

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Q20. Ads: Hypothesis Testing, Significance, and Errors

Background

Topic: Hypothesis Testing for Proportions, Significance Levels, Power, and Error Types

This question involves setting up hypotheses for a test about advertising effectiveness, understanding the impact of significance levels, and analyzing the power and error risks of the test.

Key Terms and Formulas:

• Null Hypothesis (): The proportion is 20% ().

• Alternative Hypothesis (): The proportion is greater than 20% ().

• Significance Level (): Probability of Type I error.

• Power of the Test: Probability of correctly rejecting when is true.

• Type II Error (): Probability of failing to reject when is true.

Step-by-Step Guidance

1. For part (a), write out the null and alternative hypotheses for the test about the ad's effectiveness.

2. For part (b), explain how changing the significance level from 10% to 5% affects the probability of a Type I error and the company's decision-making process.

3. For part (c), define the power of the test and explain its importance in this context.

4. For part (d), compare the power of the test at different significance levels and explain why one is higher than the other.

5. For part (e), discuss how increasing the sample size affects the risk of a Type II error and why this is the case.

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