BIG Corporation advertises that its light bulbs have a mean lifetime, , of 3200 hours. Suppose we have good reason to believe that  is different from 3200 hours and decide to do a statistical test of the claim. We choose a random sample of light bulbs manufactured by BIG and find that the mean lifetime for this sample is 3360 hours and that the sample standard deviation of the lifetimes is 650 hours. (a)What are the null hypothesis H0 and the alternative hypothesis H1 that should be used for the test? (b)	Suppose that we decide not to reject the null hypothesis. What sort of error might we be making? Type I or Type II

Let's break down the problem step by step.

### (a) Formulating the Hypotheses
We need to test whether the true mean lifetime of the light bulbs is different from the advertised mean of 3200 hours.

- **Null Hypothesis (\( H_0 \))**: This is the hypothesis that there is no effect or no difference. Here, we assume that the mean lifetime is equal to the advertised value.
  
  \[
  H_0: \mu = 3200 \text{ hours}
  \]

- **Alternative Hypothesis (\( H_1 \))**: This is what we suspect might be true. In this case, we believe that the mean lifetime might be different from the advertised value.
  
  \[
  H_1: \mu \neq 3200 \text{ hours}
  \]

This is a two-tailed test because we are interested in detecting a difference in either direction (either higher or lower than 3200 hours).

### (b) Type of Error
There are two types of errors that can be made in hypothesis testing:

1. **Type I Error**: Rejecting the null hypothesis \( H_0 \) when it is actually true. This is a false positive.

2. **Type II Error**: Failing to reject the null hypothesis \( H_0 \) when it is actually false. This is a false negative.

In this scenario, if we decide **not to reject** the null hypothesis, we could be making a **Type II Error**. This would occur if the true mean lifetime is indeed different from 3200 hours, but our test fails to detect this difference.

### Summary
- **(a)** The null hypothesis is \( H_0: \mu = 3200 \) hours, and the alternative hypothesis is \( H_1: \mu \neq 3200 \) hours.
- **(b)** If we do not reject the null hypothesis, we might be making a **Type II Error**.

Would you like to dive into the detailed steps of how to perform the hypothesis test using the sample data, or any other questions you might have?

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**Related Questions:**

1. How do you calculate the test statistic for this hypothesis test?
2. What is the significance level (\( \alpha \)) commonly used in hypothesis testing?
3. How do you interpret a p-value in hypothesis testing?
4. What are the consequences of making a Type I Error in this context?
5. What factors influence the likelihood of making a Type II Error?
6. How can the sample size affect the outcome of the hypothesis test?
7. What is the power of a test, and how is it related to Type II Error?
8. How would the conclusions change if we used a one-tailed test instead of a two-tailed test?

**Tip:** In hypothesis testing, always clearly define the hypotheses before collecting or analyzing data. This ensures that the test is focused and that the results are interpreted correctly.

Explore hypothesis testing for the mean lifetime of BIG Corporation light bulbs with a sample mean of 3360 hours, standard deviation of 650 hours, and an advertised mean of 3200 hours. Understand the null and alternative hypotheses, type I and type II errors, and interpretation of hypothesis testing results.

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