If 500 confidence intervals, each having a level of confidence of 90%, were computed for the population proportion, approximately how many of the intervals would be expected NOT to contain the population proportion?

Question

BlackTom AI · Accepted Answer

A 90% confidence level implies that, on average, 90% of constructed confidence intervals will contain the true population proportion and 10% will fail to contain it.

1. Relevant knowledge points and concepts
- Confidence level: the long-run proportion of confidence intervals (constructed the same way) that will capture the true parameter. A 90% confidence level means about 90% success, 10% failure.
- Expected count: when repeating an experiment many times, the expected number of successes (or failures) equals the number of repetitions times the probability of success (or failure).
- The statement “approximately” recognizes sampling variability; the value computed is an expectation, not a guaranteed count.

2. Complete reasoning process step by step
- There are 500 intervals constructed.
- Each interval has a 90% chance of containing the population proportion, so each has a 10% (= 1 − 0.90) chance of not containing it.
- The expected number not containing the proportion = number of intervals × probability an interval does not contain the parameter.

3. Calculations with intermediate steps
- Probability an interval does not contain the parameter = 1 − 0.90 = 0.10.
- Expected number not containing = 500 × 0.10.
- 500 × 0.10 = 50.

4. Why the correct answer is correct
- Because the confidence level directly gives the fraction of intervals expected to contain the true proportion; the complement (10%) gives the fraction expected not to contain it. Multiply that fraction by the total number of intervals to get the expected count (50).

5. Why other plausible answers would be incorrect
- 450 would be the expected number that do contain the parameter (500 × 0.90), not the number that do not.
- Any answer much larger or smaller than 50 would contradict the 10% noncoverage implied by a 90% confidence level.
- The result is an expectation (approximate), not an absolute guarantee for a particular set of 500 intervals.

6. Conclusion
- The expected number of the 500 ninety-percent confidence intervals that would NOT contain the population proportion is 50, which matches the provided answer.

If 500 confidence intervals, each having a level of confidence of 90%, were computed for the population proportion, approximately how many of the intervals would be expected NOT to contain the population proportion? 数值题

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