Math Problem Statement

Donated blood is tested for infectious diseases and other contaminants. Since most donated blood is safe, workers save time and money by testing batches of donated blood rather than testing individual samples. Workers perform a test to check if a certain toxin is present, and the entire batch is discarded if the toxin is detected. This is similar to using a null and an alternative hypothesis to determine whether to discard the batch. The hypotheses being tested could be stated as: [H_0:] The batch does not contain the toxin. [H_\text{a}:] The batch contains the toxin. Under which of the following conditions would the testers commit a Type II error? Choose 1 answer: Choose 1 answer: (Choice A) The batch does not actually contain the toxin, and they conclude it does not. A The batch does not actually contain the toxin, and they conclude it does not. (Choice B) The batch does not actually contain the toxin, and they conclude it does. B The batch does not actually contain the toxin, and they conclude it does. (Choice C) The batch actually contains the toxin, and they conclude it does. C The batch actually contains the toxin, and they conclude it does. (Choice D) The batch actually contains the toxin, and they conclude it does not. D The batch actually contains the toxin, and they conclude it does not.

Solution

A Type II error occurs when the null hypothesis (H0H_0) is not rejected, even though it is false. In this context, the null hypothesis is that "the batch does not contain the toxin."

So, a Type II error happens when:

  • The batch actually contains the toxin (which makes the null hypothesis false).
  • But the testers fail to reject the null hypothesis, meaning they incorrectly conclude the batch does not contain the toxin.

This situation is described in Choice D:

  • The batch actually contains the toxin, and they conclude it does not.

Thus, the correct answer is: (Choice D) The batch actually contains the toxin, and they conclude it does not.

Would you like further details on Type II errors, or any other aspect of hypothesis testing?

Here are some follow-up questions for you:

  1. What is the difference between a Type I error and a Type II error?
  2. How can the sample size influence the likelihood of a Type II error?
  3. What is the significance level (alpha) in hypothesis testing, and how does it relate to Type I errors?
  4. How can a researcher reduce the chances of committing a Type II error in a test?
  5. What is statistical power, and how does it relate to Type II errors?

Tip: A higher sample size generally reduces the probability of a Type II error, improving the test's power.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Type II Error
Null and Alternative Hypothesis

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12

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