Question: Rudolph Bulbs Inc. is testing the long term reliability of its holiday lights. They powered a random sample of 100 identical light strands from a recent production batch for 10,000 hours. Afterward, the number of dead bulbs in each strand was recorded. On average, 6.8 bulbs in each of the sampled strands went dead, with a standard deviation of 1.1.
Find a 99% confidence interval for the mean number of dead bulbs in a light strand from the production batch after 10,000 hours.
Round your answers to the nearest tenth.
□ < μ < □
1. **State the problem:** We want to find a 99% confidence interval for the mean number of dead bulbs $\mu$ in a light strand after 10,000 hours, based on a sample of size $n=100$ with sample mean $\bar{x}=6.8$ and sample standard deviation $s=1.1$.
2. **Formula for confidence interval:** For large samples, the confidence interval for the mean is given by:
$$\bar{x} \pm z_{\alpha/2} \times \frac{s}{\sqrt{n}}$$
where $z_{\alpha/2}$ is the critical z-value for the desired confidence level.
3. **Find the critical z-value:** For a 99% confidence level, $\alpha=0.01$, so $\alpha/2=0.005$. The critical z-value is:
$$z_{0.005} = 2.576$$
4. **Calculate the standard error:**
$$\text{SE} = \frac{s}{\sqrt{n}} = \frac{1.1}{\sqrt{100}} = \frac{1.1}{10} = 0.11$$
5. **Calculate the margin of error:**
$$\text{ME} = z_{0.005} \times \text{SE} = 2.576 \times 0.11 = 0.28336$$
6. **Construct the confidence interval:**
$$6.8 - 0.28336 < \mu < 6.8 + 0.28336$$
7. **Simplify and round to nearest tenth:**
$$6.5 < \mu < 7.1$$
**Final answer:**
$$\boxed{6.5 < \mu < 7.1}$$
This means we are 99% confident that the true mean number of dead bulbs per strand after 10,000 hours is between 6.5 and 7.1.