Let's analyze each of the given statements step by step to determine which one is correct:
1. Statement 1:
- "When constructing [tex]$95\%$[/tex] confidence intervals for 20 samples of the same size from the population, exactly [tex]$95\%$[/tex] of the samples must contain the population proportion. There must have been an error with the way our samples were chosen."
- This statement suggests that exactly [tex]$95\%$[/tex] of the samples must contain the population proportion every time we construct these intervals. However, confidence intervals are based on probabilities and there is inherent variability in sampling. Therefore, while we expect around [tex]$95\%$[/tex] of the samples to contain the population proportion, it does not have to be exactly [tex]$95\%$[/tex] every time.
- Hence, this statement is incorrect.
2. Statement 2:
- "When constructing [tex]$95\%$[/tex] confidence intervals for 20 samples of the same size from the population, the percentage of the samples that contain the population proportion should be close to [tex]$95\%$[/tex], but it may not be exactly [tex]$95\%$[/tex]."
- This statement acknowledges the probabilistic nature of confidence intervals. When we construct [tex]$95\%$[/tex] confidence intervals, we expect that about [tex]$95\%$[/tex] of the intervals will capture the true population proportion, but it might not be exactly [tex]$95\%$[/tex] due to random sampling variability.
- This is a more accurate reflection of the concept of confidence intervals.
- Hence, this statement is correct.
3. Statement 3:
- "When constructing [tex]$95\%$[/tex] confidence intervals for 20 samples of the same size from the population, at most [tex]$95\%$[/tex] of the samples will contain the population proportion."
- This statement suggests that no more than [tex]$95\%$[/tex] of the constructed intervals will contain the population proportion. This is incorrect because the point of a [tex]$95\%$[/tex] confidence interval is that we expect around [tex]$95\%$[/tex] to include the population proportion, not that [tex]$95\%$[/tex] is an upper limit.
- Hence, this statement is incorrect.
Based on this analysis, the correct statement is:
- "When constructing [tex]$95\%$[/tex] confidence intervals for 20 samples of the same size from the population, the percentage of the samples that contain the population proportion should be close to [tex]$95\%$[/tex], but it may not be exactly [tex]$95\%$[/tex]."
Thus, the final answer is:
[tex]\[
\boxed{2}
\][/tex]
