Ch. 7 - Hypothesis Testing with One Sample

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 7 - Hypothesis Testing with One Sample

Problem 7.1.54

Chapter 7, Problem 7.1.54

Getting at the Concept Explain why a level of significance of α=0 is not used.

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A level of significance (α) represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Setting α=0 would mean that there is zero probability of making a Type I error.

If α=0, it implies that we would never reject the null hypothesis, regardless of the evidence provided by the data. This would make hypothesis testing meaningless because we would always accept the null hypothesis.

In practical terms, data often contains variability and uncertainty. Even with strong evidence against the null hypothesis, setting α=0 would prevent us from making any conclusions, as no amount of evidence could lead to rejecting the null hypothesis.

The purpose of hypothesis testing is to balance the risks of Type I and Type II errors. By setting α to a small, non-zero value (e.g., 0.05), we allow for a controlled and acceptable level of Type I error while still being able to make meaningful decisions based on the data.

Therefore, a level of significance of α=0 is not used because it would eliminate the ability to reject the null hypothesis, rendering the statistical test ineffective and impractical.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Level of Significance

The level of significance, denoted as α, is the threshold for determining whether to reject the null hypothesis in hypothesis testing. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. Common values for α are 0.05 or 0.01, indicating a 5% or 1% risk of error, respectively.

Type I Error

A Type I error happens when a true null hypothesis is rejected, leading to a false positive conclusion. This error is directly related to the level of significance; a lower α reduces the likelihood of committing this error. Setting α to 0 would mean no risk of Type I error, but it would also prevent any rejection of the null hypothesis, making hypothesis testing ineffective.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether there is enough evidence to reject H0. A level of significance is crucial in this process, as it guides the decision-making regarding the hypotheses.

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Step 1: Write Hypotheses

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