Problem 41 - Basic Statistics for Business and Economics with Student

Question

A study of 25 graduates of four-year colleges by the American Banker's Association revealed the mean amount owed by a student in student loans was \(\$ 14,381 .\) The standard deviation of the sample was \(\$ 1,892 .\) Construct a 90 percent confidence interval for the population mean. Is it reasonable to conclude that the mean of the population is actually \(\$ 15,000 ?\) Tell why or why not.

Step-by-Step Solution

Answer

The 90% confidence interval is (13,735, 15,027); 15,000 is a reasonable population mean.

Step 1: Understand the Problem

We have a sample of 25 graduates with a sample mean (\( \bar{x} \)) of \( 14,381 \) and a standard deviation (\( s \)) of \( 1,892 \). We need to construct a 90% confidence interval for the population mean.

Step 2: Identify the Appropriate Formula

Since the sample size is less than 30, we'll use the t-distribution to construct the confidence interval. The formula for the confidence interval is: \[ \bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right) \] where \( n = 25 \) is the sample size.

Step 3: Find the t-Value

For a 90% confidence interval and 24 degrees of freedom (\( n-1 \)), we find the critical t-value \( t_{\alpha/2} \) from a t-table, which is approximately 1.711.

Step 4: Calculate the Standard Error

The standard error (SE) of the mean is given by: \[ SE = \frac{s}{\sqrt{n}} = \frac{1,892}{\sqrt{25}} \approx 378.4 \]

Step 5: Construct the Confidence Interval

Now, plug the values into the confidence interval formula:\[ 14,381 \pm 1.711 \times 378.4 \] This gives:\[ 14,381 \pm 646 \] So the confidence interval is approximately \( (13,735, 15,027) \).

Step 6: Interpret the Confidence Interval

Since the interval \( (13,735, 15,027) \) includes \( 15,000 \), it is reasonable to conclude that \( 15,000 \) could be the population mean. Therefore, it is reasonable to consider \( 15,000 \) as a possible value for the population mean based on this interval.

Key Concepts

T-Distribution

When we have data from a sample rather than an entire population, especially when the sample size is small (less than 30), we use the T-Distribution. This is necessary because, unlike the normal distribution, the T-Distribution is more spread out or has "fatter tails." These tails account for the added variability expected with smaller sample sizes. The T-Distribution adjusts for the fact that we estimate the population standard deviation from a small sample.
This means you are less sure about where the true population mean might fall, so the confidence interval is wider. The T-Distribution becomes increasingly similar to the normal distribution as the sample size grows. It's good to think of it as a more conservative estimate that compensates for limited data.

Sample Mean

The sample mean is an estimate of the population mean. In a sample, the mean is denoted as \[ \bar{x} \]. It's calculated by adding up all the data values and dividing by the number of observations in the sample.
In the problem, the sample mean, \[ \bar{x} \], is given as 14,381. This value is used as the midpoint of our confidence interval, serving as the best estimate of the true population mean based on the data from the sample.
Always remember that the sample mean is just an estimate and can differ from the true population mean when different samples are taken.

Standard Error

Standard Error (SE) considers how much discrepancy exists between the sample mean and the actual population mean. It represents the variability of sample means we might expect if we took multiple samples from the same population.
The formula for standard error is:\[ SE = \frac{s}{\sqrt{n}} \], where \[ s \] is the sample standard deviation and \[ n \] is the sample size. The standard error decreases as the sample size increases because there is less variability expected with larger samples.
In our study, the SE was calculated to be approximately 378.4. This value helps quantify the precision of the sample mean as an estimate of the population mean.

Degrees of Freedom

Degrees of freedom (df) is a concept that refers to the number of independent values that can vary in a calculation. In the context of calculating the T-Distribution, especially for a confidence interval, degrees of freedom is given by the formula:\[ df = n - 1 \], where \[ n \] is the sample size.
For a sample size of 25, we would have 24 degrees of freedom. This value is crucial when using a t-table to find the critical t-value needed to construct a confidence interval.Degrees of freedom are important because they affect the shape of the T-Distribution, especially with smaller sample sizes. The fewer the degrees of freedom, the thicker the tails of the T-Distribution, indicating greater uncertainty in the estimate of the population mean.