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In statistics, the standard error (SE) helps us understand how much our sample mean is expected to fluctuate. It provides a measure of the accuracy of our sample mean as an estimate of the population mean. The formula for standard error is given by:
\[ SE = \frac{\sigma}{\sqrt{n}} \]
Here, \( \sigma \) represents the population standard deviation, and \( n \) is the sample size. The formula shows that the standard error decreases as the sample size increases. This is because larger samples tend to provide more reliable estimates of the population mean. In our example, with a population standard deviation of \( 7400 \) and a sample size of \( 60 \), the standard error is calculated as approximately \( 954.77 \). This means that sample means will typically differ from the true population mean by about \( 954.77 \) on average.

Probability calculations are often made easier using the standard normal distribution, especially when dealing with sample means. The normal distribution is symmetrical, which is why a probability can often be easily determined. In statistics, this probability often relates to a particular event or range of outcomes. For example, knowing that the sample mean will be more than the population mean simply implies that there's a 50% chance, thanks to the symmetry of the normal distribution.
When calculating the probability that a sample mean falls within a certain range, such as within \\(1000 of the population mean, we use z-scores. These scores tell us how many standard errors away a value is from the mean. Once we know the z-scores for the bounds,
- For example, when calculating within \\)1000 of \$27,175, the z-scores were calculated as approximately \(-1.05\) and \(1.05\). - We can then determine the probability that the sample mean falls between these z-scores using a standard normal distribution table.
The calculated probability was approximately 0.70, indicating a 70% chance that the sample mean is within this range.

Z-scores are a key component in probability calculations involving the normal distribution. A z-score tells us the number of standard deviations a data point is from the mean. It's a way to standardize scores across different scales. This can be extremely helpful in comparing different data points to each other or assessing how unusual a data point is.
The z-score formula is:
\[ z = \frac{(X - \mu)}{SE} \]
Where:

• \( X \) is the data point, or sample mean.
• \( \mu \) is the population mean.
• \( SE \) is the standard error.

By knowing your z-score, you can easily determine the probability of a particular score using the standard normal distribution. As shown in the exercise, the z-scores helped determine the probability of a sample mean falling in a certain range, demonstrating their practical utility in statistical inference.