A sample of 169 batteries produced by a firm was found to have a mean lifespan of of 1350 hours and a standard deviation of 100hours. establish 90% confidence limits within which the mean lifespan of battries is expected to lie.

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.

A sample of 169 batteries produced by a firm was found to have a mean lifespan of 1350 hours and a standard deviation of 100 hours. Establish 90% confidence limits within which the mean lifespan of batteries is expected to lie.

Learn how to establish the 90% confidence limits for the mean lifespan of batteries, given a sample mean of 1350 hours, a standard deviation of 100 hours, and a sample size of 169. This detailed solution includes key statistical concepts like confidence intervals, normal distribution, and the Central Limit Theorem. Suitable for students in Grades 11-12 and college.

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