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A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean. If a value has a z-score of 0, it is the mean value. A positive z-score indicates the number of standard deviations an element is above the mean, whereas a negative z-score represents the element being below the mean.

For example, in our exercise, a z-score of approximately 2.33 tells us that the confidence interval's upper and lower boundaries are 2.33 standard deviations away from the mean of the sample. This is key for constructing a confidence interval, as it helps to determine how far from the sample mean the population mean \(\mu\) is likely to be with 98% confidence.

Standard error (SE) is a measure of how much discrepancy or 'error' to expect between a sample statistic, like the sample mean \(\bar{x}\), and the actual population parameter, such as the population mean \(\mu\).

The formula for SE is \(\sigma / \sqrt{n}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. The smaller the standard error, the more representative the sample mean is of the population mean. In the problem, we've calculated a standard error of 1.61. This figure is critical in determining the precision of our estimate for the population mean when we are calculating the confidence interval.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It has a bell-shaped curve and is defined by its mean and standard deviation. The total area under the curve is 1.

In the context of our exercise, the assumption that the sampled population is normally distributed permits the use of z-scores for calculating the confidence interval. It also ensures that we're applying the z-score correctly because z-scores are based on the standard normal distribution, a special case of the normal distribution where the mean is 0 and the standard deviation is 1.

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution, provided the sample size is sufficiently large (usually n>30 is considered sufficient). As the sample size increases, the shape of the distribution of the sample means will approach a normal distribution.

Given that our sample size in this exercise is more than 30, even if the population were not normally distributed, we could still apply the CLT to justify the use of normal distribution methods for estimating the population mean. This theorem is foundational for the field of inferential statistics and allows for the estimation of confidence intervals even when little is known about the actual distribution of the population.

Estimating the population mean involves using sample data to make an informed guess about the mean value of the entire population. This is often done through inference methods such as confidence intervals, which provide a range of values within which the population mean is likely to fall.

The confidence interval calculated in our exercise, [74.04, 82.36], represents the range where we can be 98% confident that the true population mean lies. This interval is created using the sample mean, standard error, and the appropriate z-score for our confidence level, demonstrating how these concepts come together to inform our estimation of the population mean.