 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 company's claim that their batteries last more than 40 hours, we can perform a one-sample t-test. Here’s the process step by step:

### Step 1: State the Hypotheses
- **Null Hypothesis (H₀):** The mean battery life is 40 hours. \[ H₀: \mu = 40 \]
- **Alternative Hypothesis (H₁):** The mean battery life is greater than 40 hours. \[ H₁: \mu > 40 \]

### Step 2: Choose the Significance Level
- The significance level (α) is 0.05.

### Step 3: Calculate the Test Statistic
We use the t-test for the mean because the sample size is small (n = 15) and the population standard deviation is unknown.

The test statistic is calculated as follows:

\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 44.9\) (sample mean)
- \(\mu_0 = 40\) (hypothesized population mean)
- \(s = 8.9\) (sample standard deviation)
- \(n = 15\) (sample size)

Substituting the values:

\[
t = \frac{44.9 - 40}{\frac{8.9}{\sqrt{15}}} = \frac{4.9}{2.298} \approx 2.13
\]

### Step 4: Determine the Degrees of Freedom and Critical Value
- Degrees of freedom (df) = \( n - 1 = 15 - 1 = 14 \).

Using a t-distribution table or calculator, the critical value of t at α = 0.05 for a one-tailed test with 14 degrees of freedom is approximately 1.761.

### Step 5: Make a Decision
- **Decision Rule:** If the calculated t-statistic is greater than the critical t-value, reject the null hypothesis.

In this case, \( t = 2.13 \) is greater than the critical value \( t_{0.05, 14} = 1.761 \).

### Step 6: Conclusion
Since the calculated t-statistic exceeds the critical t-value, we reject the null hypothesis. There is sufficient evidence at the 0.05 significance level to support the claim that the batteries last more than 40 hours.

---

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the conclusion change if the sample size were larger?
2. What is the p-value corresponding to the calculated t-statistic?
3. How would the test change if the standard deviation of the population were known?
4. What is the power of the test, given the sample size and effect size?
5. How would you conduct this test using a confidence interval instead?

**Tip:** Always check the assumptions of the t-test, such as normality of the sample distribution, especially when dealing with small sample sizes.

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.

Learn how to test a company's claim that their batteries last more than 40 hours using hypothesis testing. This detailed solution covers key concepts such as the one-sample t-test, calculating test statistics, and determining significance levels with a sample of 15 batteries. The solution is designed for students studying undergraduate statistics.

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