Problem 31 - Basic Statistics for Business & Economics

Question

Marty Rowatti recently assumed the position of director of the YMCA of South Jersey. He would like some data on how long current members of the YMCA have been members. To investigate, suppose he selects a random sample of 40 current members. The mean length of membership for the sample is 8.32 years and the standard deviation is 3.07 years. a. What is the mean of the population? b. Develop a \(90 \%\) confidence interval for the population mean. c. The previous director, in the summary report she prepared as she retired, indicated the mean length of membership was now "almost 10 years." Does the sample information substantiate this claim? Cite evidence.

Step-by-Step Solution

Answer

The population mean is estimated around 8.32 years; the 90% confidence interval is (7.50, 9.14). The sample does not support the claim of an average nearly 10 years.

Step 1: Understanding the Problem

Marty Rowatti wants to know about the average membership duration of current YMCA members. We have a sample mean of 8.32 years and a sample standard deviation of 3.07 years from 40 members.

Step 2: Identifying the Population Mean

We do not know the actual mean of the population, which is the average membership length for all YMCA members. The sample mean of 8.32 years is our best estimate for the population mean.

Step 3: Calculating the 90% Confidence Interval

To calculate the 90% confidence interval for the population mean, we use the formula for the confidence interval with a t-distribution: \[CI = \bar{x} \pm t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)\]Where \(\bar{x}\) is the sample mean, \(s\) is the standard deviation, \(n\) is the sample size, and \(t_{\alpha/2}\) is the t-value for 90% confidence with 39 degrees of freedom.

Step 4: Finding the t-value

Using a t-table or calculator for a 90% confidence interval and 39 degrees of freedom, the t-value \(t_{\alpha/2}\approx 1.685\).

Step 5: Computing the Margin of Error

Calculate the margin of error using:\[ME = 1.685 \times \left(\frac{3.07}{\sqrt{40}}\right) \approx 0.82\]

Step 6: Determining the Confidence Interval

The 90% confidence interval is calculated as:\[8.32 \pm 0.82 \rightarrow (7.50, 9.14)\]

Step 7: Evaluating the Previous Director's Claim

The previous director claimed the mean was "almost 10 years." The confidence interval (7.50, 9.14) does not include 10, suggesting the sample does not substantiate the claim.

Key Concepts

Sample Mean

The sample mean is a crucial concept when attempting to infer information about an entire population using only a subset, or sample, of that population. In this case, we are looking at the sample mean of 8.32 years regarding the membership duration of a random group of YMCA members.

• This number gives us an estimate—known as a point estimate—of what the average membership duration might be for all YMCA members.
• The true average for the whole population remains unknown, so the sample mean is our best guess.

The reliability of the sample mean depends on factors like sample size and diversity. A larger and more representative sample typically provides a better approximation of the population mean.

Standard Deviation

Standard deviation is a measure of the variability or spread in a set of data points. For our sample, it tells us how much individual membership durations deviate from the average membership duration of 8.32 years.

• A lower standard deviation implies that most membership durations are close to the sample mean.
• A higher standard deviation means there is greater variation in membership durations.

In our example, the standard deviation is 3.07 years, indicating a moderate amount of variability among the membership lengths in the sample. Understanding this variability is important when estimating how reliable the sample mean is as an estimate of the population mean.

T-distribution

The t-distribution is used when dealing with small sample sizes or when the population standard deviation is unknown, which is often the case in real-world scenarios. It is similar to the normal distribution but has thicker tails, which accounts for the additional uncertainty with small samples.

• This distribution becomes critical when constructing confidence intervals, as it helps to determine the margin within which the true population mean likely falls.

In our exercise, the t-distribution is employed to calculate the confidence interval for the population mean, using a t-value that corresponds to a 90% confidence level and 39 degrees of freedom (related to our sample size of 40). This helps ensure our interval is accurately reflecting the uncertainty in our sample.

Margin of Error

The margin of error is an essential component in statistics when it comes to constructing confidence intervals. It essentially provides a range of values around the sample mean within which we expect the true population mean to lie.

• For our example, the margin of error is calculated using the formula: \(ME = t_{\alpha/2} \times \left(\frac{s}{\sqrt{n}}\right)\).
• This formula incorporates the t-value and accounts for both sample variability (standard deviation) and sample size.

In this exercise, the margin of error is approximately 0.82, meaning that the true average YMCA membership length likely falls between 7.50 and 9.14 years, providing a more precise understanding of where the population mean might be, rather than just relying on the sample mean alone.