Identifying Type I and Type II Errors In Exercises 31–36, describe type I and type II errors for a hypothesis test of the indicated claim.

Security A campus security department publicizes that at most 25% of applicants become campus security officers.

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Understand the null hypothesis (H₀) and the alternative hypothesis (H₁). In this case, the null hypothesis (H₀) is that at most 25% of applicants become campus security officers (p ≤ 0.25). The alternative hypothesis (H₁) is that more than 25% of applicants become campus security officers (p > 0.25).

Define a Type I error. A Type I error occurs when the null hypothesis (H₀) is rejected even though it is true. In this context, a Type I error would mean concluding that more than 25% of applicants become campus security officers (rejecting H₀) when, in fact, at most 25% of applicants do become campus security officers.

Define a Type II error. A Type II error occurs when the null hypothesis (H₀) is not rejected even though it is false. In this context, a Type II error would mean failing to conclude that more than 25% of applicants become campus security officers (failing to reject H₀) when, in fact, more than 25% of applicants do become campus security officers.

Relate the errors to the practical implications. A Type I error might lead to unnecessary changes in the recruitment process based on the incorrect belief that more applicants are becoming officers than expected. A Type II error might result in missed opportunities to address issues in the recruitment process if more applicants are actually becoming officers than the department claims.

Summarize the importance of balancing Type I and Type II errors. In hypothesis testing, the significance level (α) is chosen to control the probability of a Type I error, while the power of the test (1 - β) is related to the probability of avoiding a Type II error. The choice of α and the sample size can influence the likelihood of these errors.

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Key Concepts

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

Type I Error

A Type I error occurs when a null hypothesis is incorrectly rejected when it is actually true. In the context of the campus security claim, this would mean concluding that more than 25% of applicants become security officers when, in reality, 25% or fewer do. This error can lead to unnecessary changes in policy or perception based on false evidence.

Type II Error

A Type II error happens when a null hypothesis is not rejected when it is false. For the campus security claim, this would mean failing to recognize that more than 25% of applicants become security officers when they actually do. This error can result in missed opportunities for recruitment or misjudgment of the effectiveness of the hiring process.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (the default assumption) and an alternative hypothesis (the claim being tested). In this scenario, the null hypothesis would state that 25% or fewer applicants become officers, while the alternative would suggest that the proportion is greater than 25%.

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