Problem 62 - Basic Statistics for Business and Economics with Student

Question

As part of an annual review of its accounts, a discount brokerage selects a random sample of 36 customers. Their accounts are reviewed for total account valuation. which showed a mean of \(\$ 32,000\), with a sample standard deviation of \(\$ 8,200 .\) What is a 90 percent confidence interval for the mean account valuation of the population of customers?

Step-by-Step Solution

Answer

(29,752.83, 34,247.17)

Step 1: Identify the Mean and Standard Deviation

From the problem, we note that the sample mean \( \bar{x} \) is \\(32,000 and the sample standard deviation \( s \) is \\)8,200. The sample size \( n \) is 36.

Step 2: Determine the Confidence Level and Find the Critical Value

The problem asks for a 90% confidence interval. Therefore, the confidence level \( \alpha = 0.10 \). The critical value \( z^* \) for a 90% confidence interval can be found in the standard normal (Z) distribution table or by using a calculator. For a 90% confidence level, \( z^* = 1.645 \).

Step 3: Calculate the Standard Error

The standard error (SE) of the mean can be calculated using the formula:\[ SE = \frac{s}{\sqrt{n}} \]Substituting the values, we get:\[ SE = \frac{8,200}{\sqrt{36}} = \frac{8,200}{6} = 1,366.67 \]

Step 4: Determine the Margin of Error

The margin of error (E) is calculated using the critical value and the standard error:\[ E = z^* \times SE = 1.645 \times 1,366.67 = 2,247.17 \]

Step 5: Construct the Confidence Interval

The confidence interval is given by the formula:\[ \bar{x} \pm E \]Substituting the values:\[ 32,000 \pm 2,247.17 \]This results in the interval:\[ (32,000 - 2,247.17, 32,000 + 2,247.17) = (29,752.83, 34,247.17) \]

Step 6: Interpret the Confidence Interval

We conclude that we are 90% confident that the true mean account valuation of the population of customers lies between \\(29,752.83 and \\)34,247.17.

Key Concepts

Sample Mean

In statistics, the sample mean is a way to summarize the central value of a sample from a population. It is calculated by adding all the sample values together and then dividing by the number of values in the sample. To see this in action, consider the example of 36 customers whose account valuations have been sampled. Their mean account valuation is given as \(\$32,000\).

The formula for calculating the sample mean \(\bar{x}\) is:

• \(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)

Where \(x_i\) represents each individual sample value and \(n\) is the sample size.

Understanding the sample mean helps in estimating the population mean, especially when it's impossible to review an entire population.

Standard Deviation

Standard deviation measures the amount of variability or dispersion in a set of data values. A low standard deviation means that the values tend to be close to the sample mean, while a high standard deviation indicates a wide spread of values.

In our example, the sample standard deviation is \(\\(8,200\). This is an indicator of how much the customer account valuations vary from the mean valuation of \(\\)32,000\).

The formula for standard deviation \(s\) is:

• \(s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}\)

Where \(x_i\) are the individual values, \(\bar{x}\) is the sample mean, and \(n\) is the sample size.

It's crucial to grasp the concept of standard deviation as it provides insight into the reliability and variability of your data.

Critical Value

The critical value is a point on the scale of the standard normal distribution, or Z-distribution, used to calculate the margin of error. It helps to determine the confidence level for estimating the parameter.

For a confidence interval, it specifies the extent of deviation away from the sample mean that is acceptable for the confidence level. In our case, a 90% confidence level is required, leading to a critical value \(z^*\) of 1.645.

Using a Z-table or statistical software can help find this critical value based on the desired confidence level. It is essential for constructing confidence intervals as it adds precision and context to our statistical conclusions.

Standard Error

Standard error (SE) is the estimated standard deviation of a statistical sample mean. It indicates how much the sample mean \(\bar{x}\) of the data is expected to vary from the true population mean \(\mu\). SE helps in forming the confidence interval.

The formula for standard error is:

• \(SE = \frac{s}{\sqrt{n}}\)

Where \(s\) is the sample standard deviation and \(n\) is the sample size.

In the given exercise, the SE is calculated as \(\frac{8,200}{\sqrt{36}} = 1,366.67\). This quantifies the precision of the sample mean as an estimate of the population mean. Understanding standard error aids in appreciating how much trust we can place in the sample's representation of the population.