Problem 28 - Basic Statistics for Business & Economics

Question

A state meat inspector in lowa has been given the assignment of estimating the mean net weight of packages of ground chuck labeled "3 pounds." Of course, he realizes that the weights cannot always be precisely 3 pounds. A sample of 36 packages reveals the mean weight to be 3.01 pounds, with a standard deviation of 0.03 pound. a. What is the estimated population mean? b. Determine a \(95 \%\) confidence interval for the population mean.

Step-by-Step Solution

Answer

The estimated population mean is 3.01 pounds. The 95% confidence interval is (3.0002, 3.0198) pounds.

Step 1: Understand the given data

The problem provides a sample mean weight of 3.01 pounds, a sample size of 36 packages, and a sample standard deviation of 0.03 pounds. We'll use this information to estimate the population mean and find the confidence interval.

Step 2: Estimate the population mean

Since the sample mean provides the best estimate of the population mean, the estimated population mean is simply the sample mean, which is 3.01 pounds.

Step 3: Determine the standard error

The standard error of the mean (SEM) is calculated using the formula: \( \text{SEM} = \frac{s}{\sqrt{n}} \), where \(s\) is the sample standard deviation and \(n\) is the sample size. So, \( \text{SEM} = \frac{0.03}{\sqrt{36}} = 0.005 \) pounds.

Step 4: Find the critical z-value

For a 95% confidence interval, we need to find the critical z-value, which is the z-score corresponding to the upper 2.5% of the normal distribution. This z-value is approximately 1.96.

Step 5: Calculate the margin of error

The margin of error (ME) is calculated with the formula: \( \text{ME} = z \times \text{SEM} \). Using our values, \( \text{ME} = 1.96 \times 0.005 = 0.0098 \) pounds.

Step 6: Find the confidence interval

The confidence interval is constructed as follows: \( \text{CI} = \text{Sample Mean} \pm \text{Margin of Error} \). Therefore, the 95% confidence interval is \( 3.01 \pm 0.0098 \), which gives us \( (3.0002, 3.0198) \) pounds.

Key Concepts

Population Mean

The population mean is a key concept in statistics, representing the average value of a characteristic in the entire population. In this exercise, the inspector is tasked with estimating the population mean weight of packages labeled "3 pounds." Since it is practically impossible to measure the entire population, we rely on samples.

To estimate the population mean, we use the sample mean, which is a proxy or stand-in for this unknown parameter. Given that the sample mean offers the best estimate, in our problem, the sample mean of 3.01 pounds is used to estimate the population mean. This approach assumes that our sample is representative of the entire population, which is generally considered when the samples are randomly selected.

Standard Error

The standard error (SE) quantifies the variability or spread of the sample mean estimate from the true population mean. Think of it as a measure of uncertainty, indicating how much the sample mean might vary if several samples were taken.

In our exercise, the standard error is calculated using the formula: \[ \text{SEM} = \frac{s}{\sqrt{n}} \]where \(s\) is the sample standard deviation, and \(n\) is the sample size. Here, it reflects the variation of package weights around the sample mean. The given sample standard deviation is 0.03 pounds, and the sample size is 36, resulting in a standard error of 0.005 pounds.

• This implies that if many samples were taken, the average sample mean would typically be within 0.005 pounds of the true population mean.

Sample Mean

The sample mean is the average of observed values in a sample and serves as an unbiased estimator of the population mean. It is calculated by summing all observed values and dividing by the number of observations. In the exercise, the inspector finds that the sample mean weight is 3.01 pounds from the 36 packages sampled.

This sample mean acts as a foundation for constructing a confidence interval to provide a range within which the true population mean is expected to lie.

• Sample means are crucial for making inferences about a population.
  
• They help estimate unknown population parameters based on finite, manageable observations.

Margin of Error

The margin of error (ME) represents the maximum expected difference between the sample mean and the true population mean. It accounts for sampling variability and is essential in constructing confidence intervals. It shows how precise an estimate is.

To calculate it, you multiply the critical z-value (for a given confidence level) with the standard error: \[ \text{ME} = z \times \text{SEM} \]
For a 95% confidence level, the critical z-value is approximately 1.96, leading to a margin of error of 0.0098 pounds in the exercise.

• The margin of error helps define the confidence interval, which in this example, indicates that the true population mean likely falls between 3.0002 and 3.0198 pounds.