91Ó°ÊÓ

The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that the sampling distribution of the sample mean \( \bar{x} \) will approximate a normal distribution, regardless of the shape of the population distribution, provided the sample size is sufficiently large (typically \( n \geq 30\)). This is crucial because it allows us to make inferences about the sample mean using the properties of the normal distribution. In this exercise, since a sample size of \(n = 36\) is used, the CLT ensures that \( \bar{x} \) is approximately normally distributed with a mean (\( \mu \)) of 64 and a standard error (SE) of \(3\). This theorem helps simplify complex real-world data analysis into manageable normal distribution problems.

A Z-score measures how many standard deviations an element is from the mean. It helps compare data points from different distributions. The formula to calculate the Z-score is: \[Z = \frac{X - \bar{x}}{\text{SE}} \]
In our exercise, the Z-scores are calculated to find the probabilities associated with different sample means. To find \( P(\bar{x} < 62.6) \), we calculated the Z-score as: \[Z = \frac{62.6 - 64}{3} = -0.467\] Similarly, for \( P(\bar{x} \geq 68.7)\) and \( P(59.8 < \bar{x} < 65.9)\), the Z-scores provide the probability positions. A Z-table or statistical software can then be used to determine these probabilities accurately. Understanding Z-scores allows for easier interpretation and comparison when dealing with normal distributions.

The standard error (SE) represents the standard deviation of the sampling distribution of a statistic, most commonly the mean. It indicates how much variability one can expect in the sample mean from sample to sample. SE is crucial when making inferences about the population mean from sample data.
The formula for SE is given as: \[ \text{SE} = \frac{\text{Population standard deviation (\ \sigma)}}{\text{Square root of the sample size (\ \sqrt{n} )}} \]
In our exercise, we use \( \sigma = 18\) and \( n = 36\): \[ \text{SE} = \frac{18}{\frac{\text{Square root of 36}}{}} = 3\]
This SE value then helps to calculate Z-scores and corresponding probabilities, thus quantifying the uncertainty in the estimation of the population mean. A smaller SE suggests that the sample mean is a more accurate reflection of the true population mean.