According to a report done by S & J Power, the mean lifetime of the light bulbs it manufactures is 51 months. A researcher for a consumer advocate group tests this by selecting 31 bulbs at random. For the bulbs in the sample, the mean lifetime is 52 months. It is known that the population standard deviation of the lifetimes is 7 months. Assume that the population is normally distributed. Can we conclude, at the 0.05 level of significance, that the population mean lifetime, , of light bulbs made by this manufacturer differs from 51 months?
Perform a two-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (a)	State the null H0 hypothesis H1 and the alternative hypothesis. (b)	Determine the type of test statistic to use. (c)Find the value of the test statistic. (Round to three or more decimal places.)

### (a) State the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \).

- **Null hypothesis \( H_0 \)**: The population mean lifetime of light bulbs manufactured by S & J Power is 51 months.
  \[
  H_0: \mu = 51
  \]

- **Alternative hypothesis \( H_1 \)**: The population mean lifetime of light bulbs differs from 51 months.
  \[
  H_1: \mu \neq 51
  \]
  
This is a **two-tailed test** because we are testing whether the mean lifetime differs from 51 months (it could be either greater or less than 51 months).

### (b) Determine the type of test statistic to use.

Since the population standard deviation is known and the population is normally distributed, we will use the **z-test** for the hypothesis test.

### (c) Find the value of the test statistic.

The test statistic for a z-test is calculated using the formula:
\[
z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]
where:
- \( \bar{x} \) is the sample mean,
- \( \mu \) is the population mean under the null hypothesis,
- \( \sigma \) is the population standard deviation,
- \( n \) is the sample size.

Given:
- \( \bar{x} = 52 \) months (sample mean),
- \( \mu = 51 \) months (population mean under \( H_0 \)),
- \( \sigma = 7 \) months (population standard deviation),
- \( n = 31 \) (sample size),

Substituting these values into the formula, we can compute the test statistic.

Let's calculate it.The value of the test statistic \( z \) is approximately **0.795**.

Would you like to continue with the next steps of the hypothesis test (e.g., finding the critical values, calculating the p-value, or drawing conclusions)? 

Here are some related questions you might explore:

1. What are the critical values for a two-tailed test at the 0.05 significance level?
2. How do you interpret the test statistic in the context of this problem?
3. What is the p-value associated with this z-score?
4. What would the conclusion be if the p-value is compared to the significance level?
5. How does the choice of significance level affect the decision in hypothesis testing?
6. What assumptions must be met for the z-test to be valid?
7. How would the test change if the population standard deviation were unknown?
8. What would be the test statistic if the sample mean were 50 months instead of 52?

**Tip:** Always double-check whether the assumptions of normality and known population standard deviation are satisfied before using the z-test.

Explore hypothesis testing to determine if the population mean lifetime of light bulbs made by S & J Power differs from 51 months. Includes a step-by-step solution using the z-test with given sample data.

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