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The standard error (SE) is a fundamental concept in statistics that gives insight into how much an estimated average, like the mean of a sample, is expected to vary from the true population average. It helps in understanding how accurately a sample represents the larger population. The formula for the standard error is given by:\[ SE = \frac{\sigma}{\sqrt{n}} \]
Here, \( \sigma \) represents the standard deviation of the population, and \( n \) is the sample size. This formula shows that as the sample size increases, the standard error decreases. This is because a larger sample size gives a more reliable estimate of the population mean.
In the given exercise, the standard deviation \( \sigma \) is 0.1 mg, and with a sample size of 100, the standard error is calculated to be 0.01 mg. This SE helps us estimate the variability of the sample mean, which is crucial when applying the Central Limit Theorem to find probabilities related to the sample mean.

The Z-score is a statistical measurement that tells us how many standard deviations an element is from the mean of the distribution. It is a crucial tool in determining the probability of a certain value occurring within a normal distribution.
The formula for the Z-score is:
\[ Z = \frac{X - \mu}{SE} \]
In this context, \( X \) is the sample mean you are examining, \( \mu \) is the population mean, and \( SE \) is the standard error. By calculating the Z-score, you can understand where your sample mean lies in relation to the population mean.
For the nicotine content exercise, when the sample mean \( X \) is 0.79 mg, the Z-score is -1, meaning it is one standard error below the population mean of 0.8 mg. This Z-score allows us to calculate the probability of obtaining a mean nicotine content less than 0.79 mg. Similarly, a Z-score of -3 for \( X = 0.77 \) mg indicates the sample mean is three standard errors below the population average.

A probability distribution describes how probabilities are distributed over values of a random variable. In the context of continuous data, like nicotine content, the normal distribution is often used to model the data.
Central to solving exercises like the one provided is understanding how the Central Limit Theorem impacts the probability distribution of the sample mean. This theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the original distribution's shape, provided the sample size is sufficiently large.
In our problem, we deal with a normal distribution where the sample size is 100, allowing us to apply the Central Limit Theorem effectively. By calculating the Z-scores, we could use standard normal probability tables or software to find the probabilities of the sample means being less than 0.79 mg and 0.77 mg. This understanding of the probability distribution and related tools enables us to make informed predictions about the random variable—in this case, nicotine content of cigarettes.