True or False: A 95% confidence interval for a population proportion with lower bound 0.45 and upper bound 0.51 means there is a 95% probability the population proportion is between 0.45 and 0.51.  

Verified step by step guidance

1. Understand what a 95% confidence interval represents: It means that if we were to take many samples and build a confidence interval from each sample, approximately 95% of those intervals would contain the true population proportion.
2. Recognize that the confidence interval itself is fixed once calculated, and the population proportion is a fixed but unknown value; thus, the probability statement applies to the method, not the specific interval.
3. Interpret the given interval (0.45 to 0.51) as one such interval calculated from a sample, which either contains the true population proportion or it does not, but we cannot say there is a 95% probability that the population proportion lies within this specific interval.
4. Therefore, the correct interpretation is that we are 95% confident that the interval from 0.45 to 0.51 contains the true population proportion, not that there is a 95% probability the proportion is in this interval.
5. Conclude that the statement given is false because it misinterprets the meaning of the confidence interval probability.

Key Concepts

Confidence Interval Interpretation

A confidence interval estimates a range of values likely to contain the population parameter. A 95% confidence level means that if we repeated the sampling many times, 95% of those intervals would contain the true parameter, not that there is a 95% probability the parameter lies within a single calculated interval.

Population Proportion

The population proportion is the true fraction or percentage of a population that has a particular characteristic. It is a fixed but unknown value that confidence intervals aim to estimate based on sample data.

Probability vs. Confidence

Probability refers to the likelihood of an event occurring before data is observed, while confidence relates to the reliability of an interval estimate after data is collected. Once the interval is computed, the population parameter either lies within it or not, so probability does not apply to that fixed interval.