Problem 4 - Basic Statistics for Business and Economics with Student

Question

Listed below are the 27 Nationwide Insurance agents in the Toledo, Ohio, metropolitan area. We would like to estimate the mean number of years employed with Nationwide. a. We want to select a random sample of four agents. The random numbers are: 02,59,51,25,14,29,77 \(69,\) and \(18 .\) Which dealers would be included in the sample? b. Use the table of random numbers to select your own sample of four agents. c. A sample is to consist of every seventh dealer. The number 04 is selected as the starting point. Which agents will be included in the sample?

Step-by-Step Solution

Answer

a) Agents 2, 25, 14, 18. b) Example: Agents 3, 12, 10, 5. c) Agents 4, 11, 18, 25.

Step 1: Understand the List of Agents

We have a list of 27 Nationwide agents from Toledo, Ohio, each presumably assigned a unique ID from 01 to 27.

Step 2: Use Given Random Numbers for Sample - Part a

Choose the agents corresponding to the first four random numbers that are less than or equal to 27. From the list of random numbers 02, 59, 51, 25, 14, 29, 77, 69, and 18, the numbers eligible for selection based on the list of agents are: 02, 25, 14, and 18. Therefore, the sample agents are those with IDs 2, 25, 14, and 18.

Step 3: Select Sample Using Random Numbers - Part b

To perform this step, consult a new table of random numbers. For illustrative purposes, let's assume the numbers are 03, 12, 10, and 05. Select agents corresponding to these numbers. Thus, the agents with IDs 3, 12, 10, and 5 are included in the sample.

Step 4: Systematic Sampling for Part c

Starting at agent ID 04, select every seventh agent. The IDs selected would be: 04 (starting point), 11 (4+7), 18 (11+7), and 25 (18+7). Thus, these agents are included in the sample.

Key Concepts

Systematic Sampling

Systematic sampling is an efficient way of selecting a sample from a larger population. It's particularly useful when you use a list or an ordered set of subjects or items. In systematic sampling, you choose a starting point and select every nth element in the list until you reach your desired sample size.
For example, in a situation with 27 agents and a need to pick every seventh one, you'd first randomly select a starting point. Let's say you start with agent ID 04, then you would select agent 11 (04+7), agent 18 (11+7), and finally agent 25 (18+7). This method ensures that every element in the population has an equal probability of being chosen, while also being easy to implement and understand.
You just need to be careful with your selected interval; if the list has a repeating pattern, it might introduce bias. So, always review your data before applying systematic sampling.

Random Number Table

A random number table is a tool used to ensure randomness in sample selection. It contains sequences of digits that are statistically random. These tables are often employed in statistics when random sampling is needed, especially when choosing subjects like insurance agents from a larger pool.
To use a random number table, you first assign a unique identifier to each member of your population, typically a consecutive number starting from 01. Then, you can select a starting point somewhere in your random number table, and proceed to choose numbers within your specified range. Disregard numbers outside the range because they don't correlate with any of your subjects.
This method allows for unbiased selections and is particularly reliable because it removes any subjective influence over the sampling process. The randomness from these tables helps ensure that the sample selected truly represents the diversity of the entire population.

Sample Selection

Sample selection is a crucial process in statistics that helps gather data or insights without assessing everyone in a population. This is done by choosing a subset that accurately reflects the larger group.
In practice, selecting a sample involves a few steps, like defining your population, choosing your sampling method, and then making the selection. In our exercise, we used both systematic sampling and a random number table for selection. By using different methods, you ensure that your data is reliable and unbiased. Each method has its strengths, and the choice largely depends on the available data, required accuracy, and resources.
Remember, a good sample is like a miniature version of your entire population, showing variations and characteristics in similar proportions. This ability to reflect the population accurately is what makes sample selection so important in statistics.

Mean Estimation

The ultimate goal in our example is to estimate the mean or average number of years that agents have been employed with a company. Mean estimation is a fundamental statistical concept used to describe data sets. When you select a sample from the population, you aim to calculate a sample mean, and then use this to infer the population mean. To do this accurately:

• First, gather your sample data.
• Then add together the observed values.
• Finally, divide this sum by the number of observations in your sample.

This calculation provides the sample mean. Because the sample represents the population, this sample mean will be used to make an educated guess about the population mean. Estimating the mean is essential for making predictions and decisions based on data trends. It provides a valuable summary statistic that can give you insights into the behavior or characteristics of the entire population based on your sample.