In the lecture we calculated the 95% Confidence Interval for Income as 22,138 to 22,510. Which of the following statements are correct?

95% of the data lies in the confidence interval 22,138 to 22,510
We are 95% confident the true population mean is in the confidence interval 22,138 to 22,510
Because the sample mean is in the interval, we are (95%) confident we have a good estimate
The confidence interval 22,138 to 22,510 defines a range of values for the unknown population mean that are consistent with the sample mean we observed.
We are 95% confident the sample mean lies in the interval
There is a 95% chance the true population mean is in the confidence interval 22,138 to 22,510

Question

In the lecture we calculated the 95% Confidence Interval for Income as 22,138 to 22,510. Which of the following statements are correct?

95% of the data lies in the confidence interval 22,138 to 22,510
We are 95% confident the true population mean is in the confidence interval 22,138 to 22,510
Because the sample mean is in the interval, we are (95%) confident we have a good estimate
The confidence interval 22,138 to 22,510 defines a range of values for the unknown population mean that are consistent with the sample mean we observed.
We are 95% confident the sample mean lies in the interval
There is a 95% chance the true population mean is in the confidence interval 22,138 to 22,510

BlackTom AI · Accepted Answer

A 95% confidence interval for a mean gives a range of parameter values that are consistent with the observed sample and the sampling procedure: over many repeated samples, 95% of such intervals would contain the true population mean.

1) Relevant concepts
- Confidence interval (CI): an interval computed from sample data that, using the sampling distribution, will contain the true (fixed) population parameter in 95% of repeated samples.
- The CI is about the population parameter (mean), not individual data points or the observed sample mean once computed.
- The sample mean is the center of the interval; the margin of error = half the interval width.

2) Step-by-step reasoning for these specific statements
Compute center and margin of error to show the structure:
- Center (sample mean) = (22,138 + 22,510) / 2 = 44,648 / 2 = 22,324.
- Margin of error = 22,510 − 22,324 = 186 (also 22,324 − 22,138 = 186).
This confirms the interval is sample_mean ± 186.

3) Why B and D are correct
- Statement B: "We are 95% confident the true population mean is in the confidence interval 22,138 to 22,510." This is the standard, correct interpretation: given the data and method, we have 95% confidence that the interval produced by this method includes the true mean.
- Statement D: "The confidence interval ... defines a range of values for the unknown population mean that are consistent with the sample mean we observed." Correct: the CI lists parameter values compatible with the observed sample under the 95% confidence procedure.

4) Why other individual statements are incorrect
- A ("95% of the data lies in the confidence interval"): wrong — the CI concerns the mean, not data coverage. Data points may lie well outside the CI.
- C ("Because the sample mean is in the interval, we are (95%) confident we have a good estimate"): misleading and incorrect — the sample mean being inside the interval is tautological (it’s the center); confidence refers to the procedure, not that the estimate is necessarily “good.”
- E ("We are 95% confident the sample mean lies in the interval"): false — the sample mean is a known number (22,324); confidence statements refer to the unknown population parameter.
- F ("There is a 95% chance the true population mean is in the confidence interval"): technically incorrect in frequentist terms because the true mean is fixed; probability refers to the random interval before observing data, not to the fixed parameter after observing one interval.

Conclusion: Only statements B and D are correct, so the correct choice is "B and D are correct statements," matching the provided answer.

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