Problem 12 - Basic Statistics for Business and Economics with Student

Question

Scrapper Elevator Company has 20 sales representatives who sell its product throughout the United States and Canada. The number of units sold last month by each representative is listed below. Assume these sales figures to be the population values. a. Draw a graph showing the population distribution. b. Compute the mean of the population. c. Select five random samples of 5 each. Compute the mean of each sample. Use the methods described in this chapter and \(\underline{\text { Appendix }} \mathrm{B} .6\) to determine the items to be included in the sample. d. Compare the mean of the sampling distribution of the sample means to the population mean. Would you expect the two values to be about the same? e. Draw a histogram of the sample means. Do you notice a difference in the shape of the distribution of sample means compared to the shape of the population distribution?

Step-by-Step Solution

Answer

The sample means' distribution should resemble a "normal-shaped" curve, aligning closely with the population mean due to the Central Limit Theorem.

Step 1: Graphing the Population Distribution

First, list all the sales data for the 20 representatives. Then, use statistical software or graphing tools to plot these values. The x-axis should represent the number of units sold, and the y-axis should show the frequency of representatives for those sales figures.

Step 2: Calculating the Population Mean

To find the mean of the population, sum up all the units sold by the 20 representatives and divide by 20. Use the formula: \( \text{Population Mean} = \frac{\sum x_i}{N} \) where \( x_i \) is each individual observation and \( N = 20 \).

Step 3: Selecting Random Samples

Choose five sets of 5 representatives each randomly. Use random number generators to select representatives without bias. This step can be facilitated using Appendix B.6 methods or statistical software for random sampling.

Step 4: Computing Sample Means

For each of the 5 samples, sum the units sold by that sample's representatives and divide by 5 to get the sample mean. Use the formula: \( \text{Sample Mean} = \frac{\sum x^{\text{sample}}_i}{5} \). Repeat this for all five samples.

Step 5: Comparing Mean of Sampling Distribution to Population Mean

Calculate the mean of the sample means obtained in Step 4 and compare it to the population mean from Step 2. While sampling variation can lead to slight differences, due to the Central Limit Theorem, the two averages should be close.

Step 6: Drawing a Histogram of Sample Means

Prepare data for the histogram where the x-axis represents the sample means, and the y-axis shows their frequency. Use graphing tool to create the histogram. Compare its shape to that of the original population in Step 1.

Key Concepts

Population Mean

The population mean is a measure of the central tendency that tells us the average of all values in a particular population. Knowing the population mean can help us understand the overall performance or tendency of the entire population. In the context of Scrapper Elevator Company's sales, the population mean is calculated by adding up all the units sold by the 20 sales representatives and dividing the total by 20. This gives us the average number of units sold. The formula used is:

• ext{Population Mean} = \( \frac{\sum x_i}{N} \)

where \( x_i \) represents each sales figure, and \( N \) is the total number of sales representatives, which is 20.
Understanding the population mean is essential because it provides a benchmark for evaluating how individual sales figures compare to the average.

Random Sampling

Random sampling is a crucial method in statistics that involves selecting a subset of individuals from a population to describe and make inferences about the whole population. In the exercise, five random samples of 5 sales representatives each were chosen. Random sampling ensures that every member of the population has an equal chance of being included in the sample, which helps avoid bias in data collection. This process can be executed using various tools like random number generators or statistical software. Random sampling is important because it lays the groundwork for accurate and reliable statistical inference. It allows us to estimate population parameters, like the sample mean, with greater confidence that they reflect the true characteristics of the population.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental concept in statistics that states that the distribution of sample means will tend to be normal, or bell-shaped, as the sample size becomes larger, regardless of the population's distribution. In this exercise, the CLT is observed when comparing the mean of the sample means to the population mean. According to the CLT, even if the original population distribution isn't normal, the distribution of the sample means will approximate a normal distribution, especially as more samples are included. This is why we expect the average of our sample means to be close to the population mean, despite any high variation in sales figures across the population.
The CLT helps statisticians make predictions about population parameters and can support decision-making through this understanding.

Histogram

A histogram is a graphical representation that organizes a group of data points into a user-specified range. It is particularly useful for understanding the distribution of numerical data. In this exercise, histograms were used to compare the shape of the original population distribution with the distribution of sample means. The x-axis of a histogram represents the data ranges or bins, and the y-axis indicates the frequency of data points in each bin. By comparing the histogram of the population to that of the sample means, we can visually examine differences in distribution shape due to the CLT. Whereas the population's histogram might have varied shapes, the histogram for sample means should display a more normal distribution pattern. This visual inspection is key in verifying statistical properties, such as normality and central tendency, across sampled data.