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The population mean, denoted as \( \mu \), is a critical concept in statistics. It represents the average of all the values in the entire population set. Imagine you have a huge jar filled with 10,000 marbles and you note the color of each marble, say using numbers to indicate different colors. If you want to find out what the average color number is, you'd add together all these numbers (from each marble) and then divide by the total number of marbles, that is, 10,000. The calculation might look like this: sum of marble colors divided by 10,000. This average gives you the population mean. For our exercise, the given population mean is \( \mu = 124 \). This single number tells us a lot about the entire group, like a snapshot of the whole. It's used as a reference point for further calculations like finding z-scores, which measure how far away a sample value is from this population mean.

Sample size refers to the number of observations or data points collected in a sample from the population. Here, \( n = 36 \) is the sample size. Think of it this way: if the population is a big cake, the sample is a tiny slice of it. Instead of eating the whole cake, you take a small bite to understand its taste. This bite, or slice, should be big enough to represent the whole cake. The sample size affects the reliability of the statistical findings. Larger samples generally provide more accurate estimations of the population characteristics. In our problem, the sample we've taken is 36 observations. This size is used to compute the standard error in z-score calculations, crucially influencing how precise our results are going to be. The larger the sample, the smaller the standard error, meaning you get more confidence in where the true mean lies.

The standard deviation, symbolized by \( \sigma \), shows how much variation or dispersion from the average exists in the data set. A low standard deviation means that data points are generally close to the mean, while a high standard deviation indicates that the data points are spread out over a wide range of values. Let's picture this: if the weight of students in a class varies significantly, the standard deviation will be high. Conversely, if most students weigh about the same, the standard deviation will be low. In our exercise, \( \sigma = 18 \) tells us about the spread of the population data values. When computing z-scores, this number helps us understand how much the sample mean potentially deviates from the population mean. By understanding the standard deviation, we can better grasp the significance of the spread within the data, ensuring a clearer interpretation of the z-scores.

The z-score formula measures how many standard deviations an element is from the population mean. It helps standardize individual data points for comparison. The formula is: \[ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \] Where:

• \( \bar{x} \) is the sample mean.
• \( \mu \) is the population mean.
• \( \sigma \) is the standard deviation.
• \( n \) is the sample size.

This formula involves subtracting the population mean from the sample mean, and dividing the result by the standard error (which is \( \sigma / \sqrt{n} \)). This calculation gives the number of standard deviations the sample mean is from the population mean. A z-score of 0 indicates the sample mean is exactly at the population mean. Positive z-scores signify sample means above the population mean and negative ones below. This calculation is crucial for comparing different data sets and determining outliers.