A manufacturer wishes to estimate the true mean breaking strength for a type of wire rope they produce. In a sample of 20 pieces of their wire rope, the sample mean breaking strength was found to be 36.8 kN, with an associated 95% confidence interval of 35.9 to 37.7 kN. (A kilonewton (kN) is a unit of force.) Suppose the assumptions of the procedure they used were met. Of the following options, which one is the best interpretation of that interval?

A. In repeated sampling, 95% of the 95% confidence intervals calculated in this manner will capture the sample mean breaking strength.

B. We can be 95% confident that the true mean breaking strength for this type of wire rope lies between 35.9 and 37.7 kN.

C. We can be 95% confident that the sample mean breaking strength of the 20 pieces of wire rope lies between 35.9 and 37.7 kN.

D. We can be 95% confident that the true mean breaking strength for all types of wire rope lies between 35.9 and 37.7 kN.

The best interpretation of the given interval is we can be 95% confident that the true mean breaking strength for this type of wire rope lies between 35.9 and 37.7 kN, thus the correct option is (B).

Explanation:

In a step-by-step explanation, let's break down the concept of a confidence interval:

1. Understanding Confidence Interval (CI):

- A confidence interval provides a range within which we are reasonably confident that the true population parameter lies.

- In this case, the parameter of interest is the true mean breaking strength of the wire rope.

2. Given Information:

- Sample mean breaking strength = 36.8 kN.

- Associated 95% confidence interval = 35.9 to 37.7 kN.

3. Interpretation of the Interval (Option b):

- Option b correctly interprets the confidence interval.

- It means that if we were to take many samples and calculate a 95% confidence interval for each, we would expect 95% of these intervals to include the true mean breaking strength.

- The interval (35.9 to 37.7 kN) provides a plausible range for the true mean breaking strength.

4. Comparison with Other Options:

- Option a introduces the concept of repeated sampling, which aligns with the idea of a confidence interval, but it is not as directly relevant to the given scenario.

- Options c and d focus on the sample mean breaking strength or extend the interpretation beyond the specific type of wire rope, which is not the primary focus of a confidence interval.

In conclusion, option b provides the most accurate and relevant interpretation of the 95% confidence interval in the context of estimating the true mean breaking strength, thus the correct option is (B).

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