91Ó°ÊÓ

The mean of the population, often denoted as \( \mu \), represents the average of a particular characteristic across the entire population. In this exercise, the population refers to all current members of the YMCA of South Jersey and the characteristic of interest is the length of membership.

Calculating the mean of the population is often challenging because gathering data from every single member (or unit) within the population is typically impractical due to time and resource constraints.

Instead, we draw a sample - a smaller, manageable group - and use its data to make estimates about the population mean. The sample provides a sample mean \( \overline{x} \), which serves as an unbiased estimate of the unknown population mean. In summary:

• The population mean is the true average we aim to estimate.
• In our example, the population mean is unknown, but we have a sample mean of 8.32 years.

The sample mean, denoted as \( \overline{x} \), is the average of a set of observations collected from a sample of the population. It provides an estimate of the population mean, especially when the population mean \( \mu \) is unknown or difficult to determine.

In this scenario, the director collected data from 40 random members, yielding a sample mean of 8.32 years. This number tells us, on average, how long members in this sample have been with the YMCA.

Key points about the sample mean include:

• It offers a "snapshot" of the population mean.
• While it is an estimate, it may differ due to random sample variations.
• Used in further calculations, such as developing confidence intervals to estimate the population mean.

Standard deviation is a measure of variation or spread in a set of data. For a sample, this is shown by \( s \), which tells us how much individual data points (like years of membership) deviate from the sample mean.

In Marty's case, the sample standard deviation is 3.07 years. This suggests that the individual membership durations vary, on average, by 3.07 years from the sample mean of 8.32 years.

Understanding standard deviation is important because:

• It provides insights into the consistency within the sample data.
• A larger standard deviation implies more spread, whereas a smaller one indicates that data points are closer to the mean.
• It forms a critical component of the formula for calculating the confidence interval.

When determining confidence intervals for the mean with unknown population standard deviation and smaller samples, we rely on the t-distribution. This is a probability distribution resembling the normal distribution but accounts for extra variability due to small sample sizes.

For this exercise, we have:

• Sample size of 40, leading to 39 degrees of freedom \((n-1)\).
• The t-distribution is used to determine the t-score, which adjusts our confidence interval range given the sample size.
• For our 90% confidence level, a t-score of approximately 1.685 is applied.

Using the t-distribution helps us determine:

• How precise our estimate of the population mean is likely to be.
• A credible range (confidence interval) within which the actual population mean lies with a certain level of confidence.