To establish the 90% confidence limits within which the mean lifespan of the batteries is expected to lie, we can use the formula for the confidence interval for the mean:

$\text{Confidence Interval} = \bar{x} \pm z \cdot \left(\frac{\sigma}{\sqrt{n}}\right)$

Where:

• $\bar{x}$ is the sample mean.
• $z$ is the z-score corresponding to the desired confidence level.
• $\sigma$ is the sample standard deviation.
• $n$ is the sample size.

Given Data:

• Sample mean ($\bar{x}$) = 1350 hours
• Standard deviation ($\sigma$) = 100 hours
• Sample size ($n$) = 169
• Confidence level = 90%

Step 1: Find the z-score

For a 90% confidence level, the z-score corresponds to the critical value that leaves 5% in each tail of the normal distribution. The z-score for a 90% confidence interval is approximately 1.645.

Step 2: Calculate the standard error (SE)

The standard error (SE) is calculated as:

$\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{100}{\sqrt{169}} = \frac{100}{13} \approx 7.69 \text{ hours}$

Step 3: Calculate the margin of error (ME)

The margin of error (ME) is:

$\text{ME} = z \cdot \text{SE} = 1.645 \times 7.69 \approx 12.65 \text{ hours}$

Step 4: Establish the confidence interval

Now, calculate the confidence interval as:

$\text{Confidence Interval} = 1350 \pm 12.65$

Thus, the 90% confidence interval is:

$\text{Lower Limit} = 1350 - 12.65 \approx 1337.35 \text{ hours}$ $\text{Upper Limit} = 1350 + 12.65 \approx 1362.65 \text{ hours}$

Conclusion:

The 90% confidence interval for the mean lifespan of the batteries is approximately 1337.35 hours to 1362.65 hours.

Would you like further details on this calculation or have any questions?

Here are some related questions you might find interesting:

1. How would the confidence interval change if the confidence level were 95% instead of 90%?
2. What happens to the confidence interval if the sample size increases?
3. How does a larger standard deviation affect the width of the confidence interval?
4. Can you explain the difference between a 90% and a 95% confidence interval?
5. What is the significance of the z-score in determining the confidence interval?

Tip: When interpreting confidence intervals, remember that a wider interval suggests less precision in estimating the population mean, while a narrower interval suggests greater precision.