To solve this problem, we will compute a 90% confidence interval for the population mean based on the sample data provided.

Given Information:

• Sample size ($n$) = 50
• Sample mean ($\bar{x}$) = 1570
• Sample standard deviation ($s$) = 260
• Confidence level = 90%

Step 1: Identify the critical value

Since the sample size is 50, which is greater than 30, we can use the z-distribution (normal approximation). The z-value corresponding to a 90% confidence level is found using the fact that 5% of the distribution lies in each tail, i.e., the z-value is:

$z_{\alpha/2} \approx 1.645$

Step 2: Calculate the standard error (SE)

The standard error of the sample mean is given by:

$SE = \frac{s}{\sqrt{n}} = \frac{260}{\sqrt{50}} \approx 36.77$

Step 3: Calculate the margin of error (ME)

The margin of error is the product of the critical value and the standard error:

$ME = z_{\alpha/2} \times SE = 1.645 \times 36.77 \approx 60.47$

Step 4: Compute the confidence interval

The 90% confidence interval for the population mean is given by:

$\text{CI} = \left(\bar{x} - ME, \bar{x} + ME\right) = (1570 - 60.47, 1570 + 60.47) = (1509.53, 1630.47)$

Thus, the 90% confidence interval is approximately (1509.53, 1630.47).

Step 5: Interpretation

We are 90% confident that the true mean of Big Shoulders Insurance Company's claims lies between $1509.53 and $1630.47.

Step 6: Comparison with the industry standard

The industry standard for the average claim last year was $1540. Since $1540 falls within the confidence interval (1509.53, 1630.47), there is no strong evidence to suggest that Big Shoulders Insurance Company's claims are significantly higher than the industry average.

Conclusion:

Based on the 90% confidence interval, it does not appear that Big Shoulders Insurance Company’s claims are significantly above the industry average of $1540.

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Would you like further details or have any other questions? Here are some related questions:

1. How would the confidence interval change if we had a 95% confidence level?
2. What is the effect of increasing the sample size on the width of the confidence interval?
3. How would using a t-distribution instead of a z-distribution affect the interval?
4. Could this confidence interval be used to make a decision in hypothesis testing?
5. How would the confidence interval change if the standard deviation was larger?

Tip: A narrower confidence interval provides more precise estimates but requires either a higher sample size or a lower confidence level.