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Probability is a fundamental concept in statistics that measures the likelihood that a particular event will occur. In the context of our exercise, we are interested in the probability concerning the sample mean of estimates provided by 35 economists.
When dealing with probability and estimation, we use data from samples to make inferences about a whole population. Here, we aim to find the probability that the economists' sample mean falls within a specific range of the population mean. With probability values, we generally range from 0 to 1, where 0 means the event will not occur, and 1 means it definitely will occur.
In our problem, after calculating the relevant z-scores, we used the standard normal distribution to find probabilities associated with these scores. This allowed us to determine the likelihood of the sample mean being within 1% of the population mean, which helps provide insights into the reliability of the economists' predictions.

Sampling Distribution is a critical concept when discussing the Central Limit Theorem and involves the distribution of sample means over repeated sampling. In layman's terms, it's how the means of different samples from the same population are distributed.
In our exercise, we calculated the standard deviation of the sample mean, also known as the standard error, which helps in understanding the variability of our sample statistic. This was found using the formula: \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]where \( \sigma \) is the population standard deviation, and \( n \) is the sample size. This standard error represents the average distance that any given sample mean is likely to fall from the population mean.
The Central Limit Theorem tells us that for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population's original distribution. In this exercise, this theorem allowed us to use a normal distribution to predict probabilities, which streamlined our solving process.

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values, measured in terms of standard deviations. Knowing how to calculate and interpret z-scores is crucial for comparing different data points within a dataset.
In our exercise, z-scores were calculated to determine how far the sample mean of tax savings deviates from the population mean. The formula for calculating a z-score is:\[ z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} \]where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, and \( \sigma_{\bar{x}} \) is the standard error. In the example, we found z-scores for the boundaries of our interest range (25% and 27%), helping us assess the probability of the sample mean falling within 1% of the population mean.
By converting sample means into z-scores, we were able to use the standard normal distribution table to determine these probabilities, effectively turning complex data comparison into a simpler, more interpretable task.