Testing Claim of TV Manufacturer: Hypothesis Testing and Errors
Question:
How can a consumer agency verify the claim by a television manufacturer that at least 95% of its TV sets will not need service during the first 2 years of operation?
Answer:
To verify the claim made by the television manufacturer, a consumer agency can conduct hypothesis testing to determine if the proportion of TV sets that do not need service during the first 2 years is indeed at least 95%. This involves setting up null and alternative hypotheses, performing hypothesis tests, and analyzing potential errors that can occur in the process.
Hypothesis Testing
Null Hypothesis (H0): The proportion of TV sets that do not need service during the first 2 years is at least 95%.
Alternative Hypothesis (H1): The proportion of TV sets that do not need service during the first 2 years is less than 95%.
Type I Error
Type I error occurs when the agency incorrectly rejects the null hypothesis when it is actually true. In this case, it would mean falsely accusing the manufacturer of false advertising, potentially damaging their reputation and leading to legal consequences.
Type II Error
Type II error occurs when the agency fails to reject the null hypothesis when it is actually false. This could result in the agency not taking appropriate action to protect consumers from false advertising, allowing the misleading claim to continue.
Performing Hypothesis Test
To test the claim, the agency can calculate the test statistic and compare it to the critical value. By using the one-sample proportion test with sample data from 110 purchasers, the agency can determine if the observed proportion of TV sets needing repair is inconsistent with the manufacturer's claim of at least 95% not needing service.
Conclusion
Using a significance level of α=0.05, the agency can analyze the test statistic and critical value to make a decision. If the test statistic is less than the critical value, the agency can reject the null hypothesis and conclude that the sample data are inconsistent with the claim, supporting the agency's concerns of false advertising.