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The standard error (SE) is a critical concept in inferential statistics, offering insight into how much the sample mean is expected to vary from the true population mean.
Essentially, it's a measure of the reliability of the mean calculated from a sample. The formula to determine the standard error of the mean is given by: \[ SE = \frac{\sigma}{\sqrt{n}} \]where:

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

A smaller standard error suggests that the sample mean is a more accurate reflection of the population mean. In the context of our exercise, the standard error is \( 20 \), indicating how much the sample mean spreads around the population mean. It plays a vital role in calculating confidence intervals and in hypothesis testing.
Understanding the standard error helps researchers decide on the reliability of their survey results, guiding decisions based on statistical evidence.

Calculating the sample size is vital for ensuring that a study has enough power to detect effects or differences.
In simpler terms, a properly calculated sample size means the study is likely to yield valid and reliable results. The relationship between the sample size, standard error, and population standard deviation is expressed by the formula:\[ SE = \frac{\sigma}{\sqrt{n}} \]In the provided exercise, it's important to solve this equation to find \( n \). Knowing that the standard error is \( 20 \) and the population standard deviation is \( 500 \), we can rearrange the formula:\[ 20 = \frac{500}{\sqrt{n}} \]By solving it, \( \sqrt{n} = 25 \), and squaring both sides, we find that \( n = 625 \).
A larger sample size generally leads to a smaller standard error, meaning the sample mean is more representative of the true population mean. This principle is crucial in designing experiments and interpreting data.

The normal distribution is a fundamental concept in statistics, crucial for understanding the behavior of data in many contexts. It is a continuous probability distribution that is symmetrical about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
In our exercise, we assumed that the data follows a normal distribution to calculate probabilities regarding the sample mean. Within this scenario, the data around the sample mean is expressed in terms of standard error units (SEU). Using the Z-score, which translates these units onto the standard normal curve, we identify the probability that a sample mean is within a specific range of the population mean.
Here, the range of ±1.25 SEU indicates that we are looking at the probability of the sample mean falling within the limits of ±25. Using Z-tables or standard normal distribution tools, the probability was calculated to be approximately 78.88%.
This probability is highly informative, as it quantifies the likelihood of the sample mean being close to the population mean, thus validating the accuracy and reliability of the survey's findings.