Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (a) find the sample mean
[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)

Verified step by step guidance

1. Step 1: Organize the data set provided in the table. There are 32 annual earnings values listed. Ensure all values are correctly noted for calculations.
2. Step 2: Calculate the sample mean (denoted as \( \bar{x} \)) by summing all the earnings values and dividing by the total number of data points (32). Use the formula \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) represents each individual data point and \( n \) is the sample size.
3. Step 3: Compute the sample standard deviation (denoted as \( s \)) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \). This involves finding the squared differences between each data point and the sample mean, summing them, dividing by \( n-1 \), and taking the square root.
4. Step 4: Determine the critical value (\( t \)) for the desired confidence level using a t-distribution table. The degrees of freedom (df) are \( n-1 \), where \( n \) is the sample size.
5. Step 5: Construct the confidence interval using the formula \( \bar{x} \pm t \cdot \frac{s}{\sqrt{n}} \). This involves adding and subtracting the margin of error (\( t \cdot \frac{s}{\sqrt{n}} \)) from the sample mean to find the lower and upper bounds of the interval.

Key Concepts

Sample Mean

The sample mean is the average of a set of values obtained from a sample of a population. It is calculated by summing all the values in the sample and dividing by the number of observations. The sample mean serves as a point estimate of the population mean and is a fundamental statistic used in inferential statistics.

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter with a specified level of confidence, typically 95% or 99%. It provides an estimate of uncertainty around the sample mean, indicating how much the sample mean might vary from the true population mean. The width of the interval is influenced by the sample size and variability.

Random Sampling

Random sampling is a technique used to select a subset of individuals from a larger population, ensuring that each individual has an equal chance of being chosen. This method helps to eliminate bias and allows for the generalization of results from the sample to the population. In the context of the question, the earnings data is based on a random sample of life insurance underwriters, which is crucial for valid statistical inference.