Question

An engineering student randomly sampled the train arrival times at a

nearby train station. Based on the results, the student estimates that 92%
of the trains arrived within 2 minutes of their scheduled arrival times. The
margin of error for this estimation is 3%. Which of the following is the
most appropriate conclusion about all train arrival times at the station,
based on the given estimate and margin of error?

Answer 1

To determine the most appropriate conclusion based on the given estimate and margin of error, we can use the margin of error to establish a confidence interval around the estimated proportion.

The estimated proportion is 92%, and the margin of error is 3%. This means that the true proportion of trains arriving within 2 minutes of their scheduled arrival times is likely to fall within a range that is 3% above and 3% below the estimated proportion.

So, the confidence interval for this estimate would be:

92% - 3% = 89%

92% + 3% = 95%

Now, let's evaluate the provided options:

A) It is plausible that 89% and 95% of all trains arrive within 2 minutes of their scheduled arrival times.

This option is correct. Given the margin of error, it is plausible that the true proportion of trains arriving within 2 minutes of their scheduled arrival times is between 89% and 95%.

B) Exactly 92% of all trains arrive within 5 minutes of their scheduled arrival times.

This option doesn't take the margin of error into account and is not a valid conclusion.

C) The student is 3% sure that most trains arrive within 2 minutes of their scheduled arrival times.

This option is not a valid conclusion. The margin of error does not represent the student's confidence; it represents the uncertainty in the estimate.

D) The student is 97% sure that exactly 92% of all trains arrive within 2 minutes of their scheduled arrival times.

This option incorrectly implies a high level of certainty in the exact percentage, which is not justified by the margin of error.

So, the most appropriate conclusion is option A: "It is plausible that 89% and 95% of all trains arrive within 2 minutes of their scheduled arrival times."

Answer 2

The correct answer is (A) It is plausible that 89% and 95% of all trains arrive within 2 minutes of their scheduled arrival times.

The engineering student conducted a random sampling of train arrival times at a nearby train station and estimated that 92% of the trains arrived within 2 minutes of their scheduled arrival times. However, since this estimate is based on a sample, there is a margin of error associated with it.

The margin of error in this case is 3%. This means that the true percentage of trains that arrive within 2 minutes of their scheduled arrival times could be slightly higher or lower than the estimated 92% due to random sampling variability.

To account for this margin of error, we need to consider a range within which the true percentage of trains falling within 2 minutes of their scheduled arrival times is likely to lie. In this case, we can calculate this range by subtracting the margin of error from the estimated percentage and adding it to the estimated percentage.

So, subtracting 3% from 92% gives us 89%, and adding 3% to 92% gives us 95%. Therefore, based on the given estimate and margin of error, it is plausible that between 89% and 95% of all trains arrive within 2 minutes of their scheduled arrival times.

This range takes into account the uncertainty associated with the estimate and provides a more accurate conclusion about the proportion of trains arriving within the specified time frame.