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When we talk about a population proportion, we are referring to the fraction or percentage of the population that has a specific characteristic. For example, if we wanted to find out what percentage of a city’s residents prefer public transport over private vehicles, we would be looking at the population proportion of those favoring public transport.

Calculating the population proportion involves surveying a sample of the population and then using that sample to estimate the proportion for the whole population.
Population proportion is often denoted by the symbol \( p \). For example, if 60 out of 100 surveyed people prefer public transport, the sample proportion \( \hat{p} \) would be 0.60, and we would use this to estimate the population proportion.

The confidence level tells us how sure we can be about our interval estimate; it is a measure of the reliability of the interval. Common confidence levels include 90%, 95%, and 99%.
The most widely used one is the 95% confidence level.

A 95% confidence level means we are confident that 95 out of 100 samples from the population will produce intervals that contain the true population parameter.
It doesn’t mean there's a 95% chance the population parameter lies within our specific interval.
Instead, it is about the long-term reliability of the inference process.

Interpreting confidence intervals can be tricky but is crucial for understanding statistical results.

Let’s discuss the given exercise: We have a 95% confidence interval with a lower bound of 0.45 and an upper bound of 0.51. This interval estimation means that we can be 95% confident that the true population proportion lies between 0.45 and 0.51. Not that the probability of the population proportion being between these values is 95%.
The correct way of looking at it is, if we conduct this survey multiple times and calculate confidence intervals for each instance, about 95% of those intervals would include the true population proportion.
This is why the statement, ‘there is a 95% probability that the population proportion is between 0.45 and 0.51’, is false. We should remember that the actual population proportion is either within the interval or it is not; the probability the interval will cover the population proportion (on repeated sampling) is 95%.