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The Central Limit Theorem (CLT) is one of the cornerstones of statistics. It tells us that when we take a sufficiently large sample size from a population with any distribution, the sampling distribution of the sample mean will approximate a normal distribution. Even if the original population is not normally distributed, the sample mean's distribution will be normal, as long as the sample size is large enough.

In our exercise, the sample size is 25, which is often considered adequate to apply the CLT. However, since the population is known to be normally distributed from the start, the CLT assures us that the sample mean distribution is normal too, without the need for a large sample size.

• This distribution of the sample mean will center around the population mean.
• The standard deviation of this distribution will be smaller than that of the population, specifically using the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\).

Understanding the CLT helps us assess probabilities about sample means and their deviations from the theoretical population mean. It is foundational for performing hypothesis testing and constructing confidence intervals.

The Z-Score is a statistical measurement that describes a value's relation to the mean of a group of values. In terms of the sampling distribution, the Z-score helps us determine how far and in what direction a sample mean deviates from the population mean, measured in terms of standard deviations.

To calculate the Z-score for our sample mean, we use the formula:

• \( Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}} \)

The numerator represents the deviation of the sample mean from the population mean, while the denominator standardizes this deviation by the sampling distribution's standard deviation.

For example, in our exercise, to find out if the sample mean exceeds 110, we calculate the Z-score as follows:

• \( Z = \frac{110 - 106}{2.4} \approx 1.67 \)

This Z-score represents the number of standard deviations the score of 110 is from the population mean of 106. Using a Z-score, we can consult a standard normal table to find the probability associated with this deviation.

The normal distribution, often referred to as the "bell curve," is a probability distribution that is symmetric about the mean. Most values cluster around a central peak, and probabilities for values taper off equally on both sides.

Normal distributions are vital in statistics because they represent many natural phenomena. For the purposes of inferential statistics, such as calculating probabilities from sample data, assuming a normal distribution simplifies calculations and interpretations.

• The mean, median, and mode of a normal distribution are all equal.
• The curve is symmetric around the mean.

In the scenario we're analyzing, the sampling distribution of our sample mean is normal due to the known normality of the population and the CLT. This allows us to calculate probabilities and make predictions about the sample mean, using the Z-score and normal tables.

For instance, once the sampling distribution is known to be normal, it is straightforward to calculate the probability that the mean deviates from 106 by no more than 4. This involves considering the range from 102 to 110, translating this range into Z-scores, and using the properties of the normal distribution to find probabilities.