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Understanding how to calculate the sample mean is critical in statistics as it represents the average value in a given set of data. In the context of our exercise, we're dealing with a group of 50 shoppers at a discount store. To determine if they spend more than \(\$5300\) collectively, we need to find the average spending per shopper, or the sample mean. Let's break down the process:

First, you find the total spend, which is \(\$5300\), and divide it by the number of shoppers, which is 50. This calculation gives you the sample mean (\(\mu_{sample}\)):\[ \mu_{sample} = \frac{\$5300}{50} = \$106 \].

The sample mean of \(\$106\) is a crucial benchmark. It suggests that if each shopper in the random sample of 50 spends more than \(\$106\) on average, the total expenditure will cross the \(\$5300\) mark.

The standard error (SE) provides a measure of how much variability or dispersion there is from the sample mean compared to the actual population mean. It is a pivotal concept when working with sample data, as it enables us to gauge the accuracy of the sample mean estimation. In our example:\[ SE = \frac{\sigma}{\sqrt{n}} \].

Here, \(\sigma\) represents the standard deviation of the population (\(\$30\) in our case), and \(n\) is the size of the sample, which is 50. Thus, the formula indicates that the standard error is the standard deviation divided by the square root of the sample size:\[ SE = \frac{\$30}{\sqrt{50}} \].

This computation shows us how much we would expect the average spending of a sample of 50 shoppers to deviate from the true average spending of the entire population of shoppers.

The Z-score is a statistical measurement that tells us how many standard errors a point is from the mean. In simpler terms, it quantifies the distance in terms of standard errors. The Z-score formula is:\[ Z = \frac{(\mu_{sample} - X)}{SE} \],

where \(\mu_{sample}\) is the sample mean, X is the mean of the population, and SE is the standard error. Applying this formula to the exercise, with a population mean of \(\$100\), a sample mean of \(\$106\), and the previously calculated SE, we get:\[ Z = \frac{(106 - 100)}{SE} \].

After calculating the SE, you would then use this Z-score to find out how unusual or common the sample mean is in relation to the population mean.

When we talk about probability distributions, we're referring to the mathematical function that provides the probabilities of occurrence of different possible outcomes. In this scenario, we're interested in the normal distribution because the money spent by customers can be assumed to follow a bell-shaped curve with most of the data clustering around the mean.
To find the probability that our sample mean is greater than \(\$106\), we look at the right tail of the normal distribution. Using the calculated Z-score, we consult standard Z-tables which provide the areas under the curve. By finding out the area to the left of our Z-score and subtracting it from 1, we're left with the area to the right, which represents the probability of the sample mean exceeding \(\$106\).

This method allows us to use the properties of the normal distribution to make inferences about our sample in relation to the entire customer population and answer questions about spending habits with a degree of certainty.