Problem 7 - Basic Statistics for Business and Economics with Student

Question

A population consists of the following five values: 12 , \(12,14,15,\) and 20 a. List all samples of size \(3,\) and compute the mean of each sample. b. Compute the mean of the distribution of sample means and the population mean. Compare the two values. c. Compare the dispersion in the population with that of the sample means.

Step-by-Step Solution

Answer

Sample means are more clustered around the population mean. Dispersion in sample means is less than in population.

Step 1: Identify All Possible Samples

List all combinations of sampling three values from the population \( \{12, 12, 14, 15, 20\} \). Use combinations without replacement: \( \{12, 12, 14\}, \{12, 12, 15\}, \{12, 12, 20\}, \{12, 14, 15\}, \{12, 14, 20\}, \{12, 15, 20\}, \{12, 14, 15\}, \{12, 14, 20\}, \{12, 15, 20\}, \{14, 15, 20\} \).

Step 2: Calculate Sample Means

Calculate the mean for each sample identified in Step 1. For example, for sample \( \{ 12, 12, 14 \} \): Mean = \((12+12+14)/3 = 12.67\). Repeat for all samples.

Step 3: Compute the Mean of Sample Means

Find the average of all sample means calculated in Step 2. Sum up all the sample means and divide by the number of samples, which is 10.

Step 4: Calculate the Population Mean

Find the mean of the entire population. Add all the population values together and divide by 5: \((12 + 12 + 14 + 15 + 20)/5\).

Step 5: Compare Means of Distribution of Sample Means and Population

Compare the mean of the sample means (from Step 3) against the population mean (from Step 4). Discuss any discrepancies or similarities.

Step 6: Calculate Dispersion in Population

Calculate the variance of the population by finding the mean of the squared deviations from the mean. Use the formula for variance: \( \sigma^2 = \sum (x_i - \mu)^2 / N \).

Step 7: Calculate Dispersion in Sample Means

Calculate the variance of the sample means calculated in Step 3. This involves computing the mean of squared deviations of sample means from the mean of sample means.

Step 8: Compare Dispersion

Compare the variance obtained from Step 6 with that from Step 7 and discuss the dispersion differences between the population and the sample means.

Key Concepts

Sampling Distribution

When you draw a sample from a population and calculate a statistic, such as the sample mean, you can look at how that statistic varies if you repeat this process multiple times. This variation is depicted in what we call a sampling distribution. Essentially, it represents the distribution of that statistic calculated from various possible samples drawn from the same population.

• Imagine collecting multiple samples of size 3 from the population values: 12, 12, 14, 15, and 20.
• After listing all combinations and computing their means, you form the sampling distribution of the sample mean.

The beauty of the sampling distribution is that it allows statisticians to understand the behavior of the sample mean, which is crucial when making inferences about the entire population from a sample.

Population Mean

The population mean is the average of all individual values in a population, providing a central value for the data. It's a fixed parameter, often denoted by the Greek letter \(\mu\). To find it, you sum all the values and divide by the number of values.

• In our case, the population consists of 12, 12, 14, 15, and 20.
• Add these numbers to get 73, then divide by the count of 5 to obtain a population mean of 14.6.

Understanding the population mean helps in comparing it with sample data, providing insights into how well the sample represents the population.

Variance

Variance measures how far a set of numbers is spread out from their average value. In statistics, it helps us understand the dispersion of scores in a population or sample. For a population, variance is denoted as \(\sigma^2\), calculated by averaging the squared deviations from the mean.

• If our population is 12, 12, 14, 15, and 20, calculate each value's deviation from the mean (14.6), square them, and then average those squares.
• This gives us the variance for the population, a critical measure indicating how much individual numbers deviate from the mean on average.

Variance is foundational in statistics because it offers a comprehensive view of how data points differ from the population mean.

Dispersion

Dispersion refers to the spread of a dataset and shows whether the data points are tightly clustered or widely spread out.

• In evaluating the dispersion of the population, we compute the variance, which indicates how much the numbers in your dataset differ from the mean.
• Similarly, with the sample means, dispersion is assessed by calculating the variance of these means, which shows how much sample means vary from the mean of the sample means.

Comparing the dispersion values helps in understanding how variable the data is within the population, as well as across different samples. This comparison is key in statistical analyses, particularly when deciding whether the sample means are a good representation of the population mean.