Problem 56 - Basic Statistics for Business and Economics with Student

Question

A random sample of 25 people employed by the Florida Department of Transportation earned an average wage (including benefits) of \(\$ 65.00\) per hour. The sample standard deviation was \(\$ 6.25\) per hour. a. What is the population mean? What is the best estimate of the population mean? b. Develop a 99 percent confidence interval for the population mean wage (including benefits) for these employees. c. How large a sample is needed to assess the population mean with an allowable error of \(\$ 1.00\) at 95 percent confidence?

Step-by-Step Solution

Answer

a) The population mean is estimated to be \$65.00. b) The 99% confidence interval is (61.78, 68.22). c) A sample size of 151 is needed.

Step 1: Understanding the Sample Mean and Estimate of the Population Mean

In statistics, the sample mean is used as an estimate for the population mean, particularly when the sample is randomly selected. Here, the sample mean wage is given as \(\$65.00\) per hour. Since the sample is a representation of the population, the sample mean is the best estimate of the population mean until further statistical analysis is done.

Step 2: Calculating the 99% Confidence Interval

To calculate the 99% confidence interval for the population mean, we use the formula: \[ \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right) \] where \(\bar{x} = 65.00\), \(s = 6.25\), and \(n = 25\). The \(z\)-score for a 99% confidence level is approximately 2.576 (from standard normal distribution tables). Compute the margin of error: \[ 2.576 \left( \frac{6.25}{\sqrt{25}} \right) = 2.576 \times 1.25 = 3.22 \]. Therefore, the confidence interval is \(65.00 \pm 3.22\), or (61.78, 68.22).

Step 3: Calculating the Sample Size for 95% Confidence with $1.00 Error

To determine the required sample size \(n\) with a margin of error of \(\$1.00\) at 95% confidence, use the formula: \[ n = \left( \frac{z \times s}{E} \right)^2 \] where the \(z\)-score for 95% confidence is 1.96, \(s = 6.25\), and allowable error \(E = 1.00\). Substitute the values: \[ n = \left( \frac{1.96 \times 6.25}{1.00} \right)^2 = \left( 12.25 \right)^2 = 150.06 \]. Rounding up, the sample size needed is 151.

Key Concepts

Sample Mean

In statistics, the sample mean is a crucial concept when making inferences about a population. It acts as the central value around which data points in a sample are distributed. In simpler terms, it's the average value of a set of numbers taken from a larger population. The formula to calculate the sample mean is straightforward: \[ \bar{x} = \frac{\sum x}{n} \] where \( \sum x \) is the sum of all sample observations, and \( n \) is the number of observations in the sample. Why is the sample mean important?

• It's used as an unbiased estimator of the population mean when our sample is randomly selected.
• In many statistical analyses, it's the fundamental measure of central tendency.

For instance, in the provided exercise, the sample mean wage of \$65.00 per hour gives us a preliminary idea about the average wage of all employees in the Department of Transportation. Until further calculations, the sample mean serves as our best estimate of the true population mean.

Population Mean

The population mean is a statistical measure that represents the average of a set of numerical data from an entire population. Unlike the sample mean, the population mean (often denoted as \( \mu \)) is sometimes hard to determine directly unless you have access to data from every member of the population. When we can't gather data from every individual, we use the sample's data to infer information about the population mean. The central limit theorem supports this process by implying that the sample mean will approximate the population mean as the sample size becomes larger.Key points about the population mean:

• It's a constant value and unaffected by random sampling variability.
• Calculation directly requires data from every member of the population, which is often impractical.

In our exercise example, while the true population mean wage is unknown, the sample mean of \$65.00/hour is used as an initial estimate for it, especially when constructing confidence intervals.

Sample Size Calculation

Sample size calculation is a critical step when planning data collection for statistical analysis. It helps to ensure that the sample accurately reflects the population from which it is drawn. The formula used to compute the necessary sample size depends on the desired accuracy and confidence level of the estimate. Here's the basic formula for determining sample size when estimating a population mean with a specified margin of error \( E \): \[ n = \left( \frac{z \times s}{E} \right)^2 \] where:

• \( z \) is the z-score corresponding to the desired level of confidence.
• \( s \) is the sample standard deviation.
• \( E \) is the allowable error or margin of error.

For example, if we want to estimate the population mean wage with a margin of error of \\(1.00 at a 95% confidence level, using the standard deviation of \\)6.25, the necessary sample size is calculated to be 151. This ensures that the estimate is precise enough for our analysis and decision-making processes.