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The Central Limit Theorem (CLT) is a fundamental concept in statistics. It states that the distribution of the sample means will approximate a normal distribution as the sample size gets larger, regardless of the population's distribution shape. This approximation to normality becomes quite accurate when the sample size is greater than or equal to 30. In the context of Jan's All You Can Eat Restaurant,

• The expenses per customer have a skewed distribution, indicating that not all data points are spread evenly around the mean.
• However, when considering the average expense for a sample of 100 customers, CLT allows us to presume this sampling distribution of the average costs is normally distributed.

The significance of the Central Limit Theorem here is that it justifies using normal distribution techniques to estimate probabilities and make inferences, even if the original data isn't normally distributed. This is why we can accurately find the sampling distribution of the average expense and use such techniques to determine profitability.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In simpler terms, it tells us how much the values deviate from the mean value. A small standard deviation indicates that the data points tend to be very close to the mean, while a large standard deviation indicates that the data points are spread out over a large range of values. For Jan's restaurant:

• The population standard deviation of expense per customer is given as $3.
• To find the standard deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size.
• This calculation is \[ \sigma_{\bar{x}} = \frac{3}{\sqrt{100}} = 0.3 \]

This results in a smaller standard deviation for the sampling distribution than the original data's standard deviation. It indicates that the averages of our random samples are less spread out around the mean than individual observations.

A Z-score signifies how many standard deviations an element is from the mean. Calculating a Z-score helps us understand where a particular value falls within the distribution. The formula for a Z-score is:\[ Z = \frac{X - \mu}{\sigma} \]where:

• \( X \) is the value we're examining,
• \( \mu \) is the mean, and
• \( \sigma \) is the standard deviation.

In Jan's restaurant scenario, when evaluating if the restaurant makes a profit, we calculate:\[ Z = \frac{8.95 - 8.20}{0.3} = 2.5 \]A Z-score of 2.5 tells us that the observed sample mean (\( 8.95 \)) is 2.5 standard deviations above the population mean of expense (\( 8.20 \)). This information is crucial because:

• It allows us to use normal distribution tables to find the probability that the sample mean is less than \( 8.95 \), ensuring profitability.
• Looking up a Z-score of 2.5 in a standard normal distribution table, we find a corresponding probability of 0.9938, indicating a high likelihood (99.38%) of making a profit on that specific day.