A company wants to test the claim that their batteries last more than 40 hours. Using a simple random sample of 15 batteries yielded a mean of 44.9 hours, with a standard deviation of 8.9 hours. Test this claim using a significance level of 0.05.

To test the claim that the batteries last more than 40 hours, we can use a one-sample T-test for the mean. Here's how to approach the problem:

### Given Data:
- Population mean (\(\mu\)): 40 hours (This is the value being tested against)
- Sample mean (\(\bar{x}\)): 44.9 hours
- Sample standard deviation (\(s\)): 8.9 hours
- Sample size (\(n\)): 15
- Significance level (\(\alpha\)): 0.05

### Step 1: State the Hypotheses

- **Null hypothesis (\(H_0\))**: The true mean battery life is 40 hours.
  \[
  H_0: \mu = 40 \text{ hours}
  \]
- **Alternative hypothesis (\(H_1\))**: The true mean battery life is greater than 40 hours.
  \[
  H_1: \mu > 40 \text{ hours}
  \]

This is a **one-tailed** test because the claim is that the mean is greater than 40 hours.

### Step 2: Calculate the Test Statistic

The formula for the T-test statistic is:
\[
t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
\]

Substituting the given values:
\[
t = \frac{44.9 - 40}{\frac{8.9}{\sqrt{15}}}
\]

Let's calculate the value:

\[
t = \frac{4.9}{\frac{8.9}{\sqrt{15}}} \approx \frac{4.9}{2.299} \approx 2.13
\]

### Step 3: Determine the Degrees of Freedom and Critical Value

- Degrees of freedom (\(df\)) = \(n - 1 = 15 - 1 = 14\)
- For a one-tailed test at the 0.05 significance level, the critical value of \(t\) for \(df = 14\) is approximately \(t_{\text{critical}} = 1.761\) (from the T-distribution table).

### Step 4: Compare the Test Statistic to the Critical Value

- If \(t > t_{\text{critical}}\), we reject the null hypothesis.
- Here, \(t = 2.13\) and \(t_{\text{critical}} = 1.761\).

Since \(2.13\) is greater than \(1.761\), we reject the null hypothesis.

### Conclusion:
There is sufficient evidence at the 0.05 significance level to support the claim that the company's batteries last more than 40 hours.

Would you like to explore more details or have any questions?

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**Relative Questions:**
1. How would the test result change if the sample standard deviation was 10 hours?
2. What does the p-value represent in this context, and how does it relate to the significance level?
3. How does a one-tailed test differ from a two-tailed test in hypothesis testing?
4. What would the conclusion be if the sample mean was exactly 40 hours?
5. How would increasing the sample size impact the test statistic and conclusion?

**Tip:** In hypothesis testing, ensure that the direction of the test (one-tailed vs. two-tailed) aligns with the claim you're investigating. This helps in correctly interpreting the results.

Explore how to test a claim that battery life is greater than 40 hours using a one-sample T-test. Includes step-by-step calculations with a sample mean of 44.9 hours, sample standard deviation of 8.9 hours, and a significance level of 0.05.

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