Correct option is B
(2)They are a way of measuring the precision of the estimate of the mean.
Option 1: They are limits between which, in the long run, 95% of observations fall.
- Incorrect.
- This describes the concept of a 95% prediction interval, not a confidence interval. A prediction interval gives the range where we expect individual observations to fall, not the estimated mean. A confidence interval is related to the estimate of the mean, not individual data points.
Option 2: They are a way of measuring the precision of the estimate of the mean.
- Correct.
- A 95% confidence interval shows the range within which the true population mean is expected to fall with 95% confidence. It measures the precision of the sample mean as an estimate of the population mean. A narrower confidence interval indicates greater precision in estimating the mean.
Option 3: They are limits within which, the sample mean falls with probability 0.95.
- Incorrect.
- This is a common misconception. The sample mean is a fixed value, and it is not random once it is calculated from the sample. The 95% confidence interval refers to where the true population mean is likely to fall, not the sample mean. In other words, the sample mean does not change after it's computed, so it does not "fall" within a range with probability.
Option 4: They are a way of measuring the variability of a set of observations.
- Incorrect.
- This statement is related to the variance or standard deviation of the observations, not a confidence interval. The confidence interval is about estimating the population mean, not measuring the variability of the data.
