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The sample proportion, denoted as \(\hat{p}\), plays a critical role in statistics, particularly when working with categorical data. It represents the proportion of successes or specific outcomes of interest observed in a sample. For instance, if you wish to know the percentage of students who passed an exam in a certain class, the sample proportion \(\hat{p}\) is equal to the number of students who passed divided by the total number of students in the class.

When computing a confidence interval, the sample proportion provides an estimate of the true proportion, or population proportion, based on the sampled data. It's crucial to understand that \(\hat{p}\) is just an estimate and may vary from sample to sample; that's where confidence intervals come into play. They allow us to establish a range of plausible values for the population proportion, with a certain level of certainty, based on the sample proportion.

In your given exercise, the sample proportion \(\hat{p}\) is 0.85, indicating that, for example, 85% of a sample of size 120 exhibited the characteristic of interest. This \(\hat{p}\) is the cornerstone for subsequent calculations of the confidence interval.

The sample size, denoted as \(n\), is simply the number of observations included in the sample. It's a fundamental concept in statistics because it affects the precision of our estimates. Generally speaking, larger sample sizes tend to produce more reliable estimates of a population parameter, such as the population proportion.

Why is sample size so significant when calculating confidence intervals? As you increase your sample size, the margin of error within your confidence interval tends to decrease. This means that you can be more confident that your interval includes the true population parameter. The math behind this involves the calculation of standard error, as the standard error decreases with an increasing sample size.

In the context of your exercise, with a sample size of 120, we have enough information to compute the standard error and the confidence interval for the population proportion. Remember, a smaller sample size might widen the interval due to more considerable variability, while a larger sample size could tighten it.

The standard normal distribution, often referred to as the Z-distribution, is a special case of the normal distribution. It's defined to have a mean of zero and a standard deviation of one. This distribution is used extensively in statistics, particularly in the realm of inferential statistics, where it helps determine how far our sample statistic deviates from what we would expect if the null hypothesis were true.

In the calculation of confidence intervals for the sample proportion, we use the Z-distribution to find critical values (often denoted as \(Z*\)). These critical values serve as multipliers for the standard error in the confidence interval formula to capture the desired level of confidence. For instance, if you want to be 90% confident, you identify the \(Z*\) value that holds 90% of the distribution's probability in the center, and that value serves as the cut-off for your interval.

In the given exercise solution, the critical value \(Z*\) of 1.645 corresponds to a 90% level of confidence. It specifies how many standard errors to go above and below the sample proportion to obtain the interval. This is crucial, as it ensures that the interval you calculate appropriately reflects the confidence level you desire to maintain in your estimate.