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The sampling distribution of \( \bar{x} \) refers to the probability distribution of the sample mean from a set of samples taken from the same population. According to the Central Limit Theorem, the sampling distribution of \( \bar{x} \) tends to be normal if the sample size is sufficiently large, even if the population distribution is not normal. For our case, where \( n = 36 \), the theorem applies, and we can state that the sampling distribution of \( \bar{x} \) is approximately normal.

The mean of the sampling distribution (\( \text{μ}_{\bar{x}} \)) is equal to the population mean (\( \text{μ}=64 \)), and the standard deviation of the sampling distribution (also known as the standard error) is calculated using the formula: \(\text{σ}_{\bar{x}} = \frac{\text{σ}}{\text{√n}} \) which in our example simplifies to \( \frac{18}{6} = 3 \). Therefore, \( \text{μ}_{\bar{x}} = 64 \) and \(\text{σ}_{\bar{x}} = 3 \).

A Z-score quantifies the number of standard deviations a data point is from the mean of the sampling distribution. It helps in standardizing different data points so we can compare them.

For instance, to find the probability \( P(\bar{x}<62.6) \), we compute the Z-score using the formula: \( z = \frac{x - μ_{\bar{x}}}{σ_{\bar{x}}} \). Substituting into the equation, \( z = \frac{62.6 - 64}{3} = -0.467 \). This Z-score tells us how far away 62.6 is from the average sample mean in standard deviation units.

The standard error (SE) measures an estimate of the standard deviation of the sample mean's sampling distribution. It provides an understanding of how much variability we can expect in the sample mean from sample to sample.

From our example, with \( n = 36 \) and a population standard deviation of \( \text{σ} = 18 \), the SE is calculated as \( \frac{\text{σ}}{\text{√n}} = \frac{18}{6} = 3 \). The smaller the SE, the more accurate our sample mean is as an estimate of the population mean.

Probability calculations allow us to find the likelihood of a certain sample mean occurring within the sampling distribution.

For instance, to find \( P(\bar{x} \text{ < } 62.6) \), we compute the Z-score and then find the corresponding probability using a Z-table. Given \( z = -0.467 \), we find the probability of \( P(z < -0.467) \) which is approximately 0.3192.

Similarly, for \( P(\bar{x} \text{ ≥ } 68.7) \), the process involves computing \( z = 1.567 \), finding the area to the left from Z-tables and subtracting it from 1 to get the result: \( P(\bar{x} \text{ ≥ } 68.7) = 0.0571 \).

The population mean (\( \text{μ} \)) is the average of all values in the population. It is a fixed value and doesn't change. In our example, the population mean is given as 64.

In sampling, we use the population mean to compare with sample means. The goal is often to understand how sample results relate to the overall population.

When computing probabilities or Z-scores, we use this population mean as our benchmark value.

The standard deviation (\text{σ}) measures how spread out the values in a population are. It gives insights into data variability.

In our case, the population standard deviation is 18. When we take samples, we often calculate the standard error (SE), which is the standard deviation of the sampling distribution. For our given sample size, we use \( \text{σ}_{\bar{x}} = \frac{\text{σ}}{\text{√n}} = \frac{18}{6} = 3 \).

Standard deviation helps in understanding the dispersion of data and is a key component in calculating Z-scores and probabilities in a sampling distribution.