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The Standard Error of the Mean (SEM) is a key statistical concept that helps us understand the variability of sample means around the population mean. When you take a sample from a population, each sample mean can slightly differ from the true population mean. The SEM measures how much these sample means tend to vary.To calculate the SEM, use the formula: \[ SEM = \frac{\sigma}{\sqrt{n}} \] where:

• \( \sigma \) is the standard deviation of the population
• \( n \) is the size of the sample

As the sample size increases, the SEM decreases, indicating that the sample mean is likely to be closer to the population mean. In practical terms, for a large enough sample size, like 50 in this exercise, the SEM becomes small and thus makes our estimated mean more reliable.
For our exercise, with a population standard deviation of \( \$250 \) and a sample size of 50, the SEM is calculated as approximately \( 35.36 \). This value quantifies the expected deviation of the sample mean from the population mean, thus enabling us to conduct further probability calculations.

In statistics, the Normal Distribution is a crucial concept used for probability and inferential statistics. It's a symmetric, bell-shaped distribution where most of the data points tend to cluster around the mean.The significance of the Normal Distribution becomes apparent when applying the Central Limit Theorem (CLT), which allows us to use normal distribution properties even when the underlying data is not normally distributed.
The CLT tells us that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough (usually \( n \geq 30 \)).In this particular exercise, the original distribution of monthly rents is positively skewed. Despite this skewness, because we are working with a sample size of 50, the CLT assures us that we can approximate the sampling distribution of the mean as normal. This justifies the use of the normal z-score in our next steps to calculate probability.

A Z-score, or standard score, indicates how many standard deviations an element is from the mean. It is used to determine the probability of a statistic occurring within a normal distribution and to compare different data points in a dataset.The formula for the Z-score is:\[ z = \frac{\bar{x} - \mu}{SEM} \]where:

• \( \bar{x} \) is the sample mean
• \( \mu \) is the population mean
• SEM is the standard error of the mean

In our exercise, the Z-score tells us how unusual or usual our sample mean is in the context of the population mean. A Z-score of -7.07, as calculated, suggests that the sample mean of \( \\(1950 \) is extremely rare and significantly lower than the mean. In practical terms, this means the probability of observing such a low sample mean is nearly zero. However, since we need the probability of finding the mean to be at least \( \\)1950 \), we look at the complementary probability given that Z-scores take into account cumulative probabilities.