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A population parameter is a crucial concept in statistics. It refers to a numerical summary of a characteristic for the entire population. Imagine you want to know some true characteristic about every match in the FA Premier League where the outcome isn't a tie. That's your population, consisting of all such matches.
For instance, in the given exercise, the population parameter of interest is the true proportion of matches where the home team wins. This is represented by the symbol \( p \). It's essential to understand that while the value of \( p \) is fixed, it is often unknown because collecting data from each member of the population can be practically impossible.
Instead, statisticians use methods to estimate this parameter using samples. In our case, the sample consists of 120 matches, out of which we observed 70 victories by the home team. By studying these samples, we aim to make inferences about the larger population.

The sample we collect from a population allows us to calculate what's known as a sample proportion. This is often denoted by \( \hat{p} \), where the hat symbol indicates it's an estimate of the true proportion \( p \). In our scenario, \( \hat{p} \) represents the proportion of matches won by the home team from our sample of 120 matches.
To determine \( \hat{p} \), we divide the number of successes in our sample (victories by the home team) by the total number of observations (the matches sampled). So, \( \hat{p} = \frac{70}{120} = 0.583 \). This value of 0.583 tells us that in about 58.3% of matches, the home team was victorious in our sample.
Keep in mind, this sample proportion is a way to approximate the true underlying population parameter and provides a foundation for further statistical analysis, like constructing confidence intervals to make more precise inferences.

The process of constructing confidence intervals allows us to determine statistical evidence regarding a hypothesis. When we talk about statistical evidence, we're talking about how likely it is that our observations are reflective of the true state of the population.
By calculating confidence intervals, such as the 90% or 99% intervals done in this exercise, we're essentially saying, "Here's a range where we expect the true population parameter to be." The intervals provide boundaries based on our sample data, helping us infer about the whole population.
If a confidence interval for a parameter, like the proportion of home wins, doesn't include certain critical values (like 0.5 in our case), it provides evidence of a real effect or trend. Since neither the 90% nor the 99% confidence interval includes 0.5, this suggests that the home teams win more often than 50% of the time, giving us statistical evidence to suggest a home field advantage exists.