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.


Phone Repairs A cell phone repair shop advertises that the mean cost of repairing a phone screen is less than \$120.

Verified step by step guidance

1. Understand the null hypothesis (H₀) and the alternative hypothesis (Hₐ): The null hypothesis (H₀) is that the mean cost of repairing a phone screen is \$120 or more (H₀: μ ≥ 120). The alternative hypothesis (Hₐ) is that the mean cost of repairing a phone screen is less than \$120 (Hₐ: μ < 120).
2. 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, it means concluding that the mean cost of repairing a phone screen is less than \$120 when, in reality, it is \$120 or more.
3. 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, it means failing to conclude that the mean cost of repairing a phone screen is less than \$120 when, in reality, it is less than \$120.
4. Relate the errors to the context of the problem: A Type I error might lead the repair shop to advertise a lower mean repair cost than is actually true, potentially misleading customers. A Type II error might result in the shop not advertising the lower mean repair cost, missing an opportunity to attract more customers.
5. 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 used to minimize the probability of a Type II error. The shop must carefully consider these trade-offs when conducting the test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (H1), which indicates the presence of an effect or difference. In this context, the null hypothesis would state that the mean cost of repairing a phone screen is $120 or more, while the alternative hypothesis would claim it is less than $120.

Type I Error

A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. In the context of the phone repair shop's claim, this would mean concluding that the mean cost of repairing a phone screen is less than $120 when, in reality, it is $120 or more. This type of error can lead to false claims and potentially mislead customers about the pricing of services.

Type II Error

A Type II error happens when the null hypothesis is not rejected when it is false. In this scenario, it would mean failing to recognize that the mean cost of repairing a phone screen is indeed less than $120 when it actually is. This error can result in missed opportunities for the repair shop to promote their services effectively, as they would not be able to substantiate their claim with statistical evidence.