Question

Suppose a telephone company executive wishes to select a random sample of \(n=20\) out of 7000 customers for a survey of customer attitudes concerning service. If the customers are numbered for identification purposes, indicate the customers whom you will include in your sample. Use the random number table and explain how you selected your sample.

Short answer

Answer: 20 customers

Step-by-step solution

Review the random number table

Obtain a random number table. A random number table is a list of numbers, where each number has an equal probability of being chosen. The random number table helps us generate random numbers to select customer IDs in our sample.

Determine the number of digits in the customer IDs

Since there are 7000 customers, the customer IDs range from 1 to 7000. In order to select customers using the random number table, we will consider customer IDs as four-digit numbers (with leading zeros for IDs less than 1000). For example, customer ID 3 would be represented as 0003, and customer ID 6789 would be 6789.

Generate random customer IDs

Using the random number table, select 20 sets of four-digit numbers. Each four-digit number will correspond to a customer ID. Ensure that the selected numbers are within the range of 0001 to 7000. For example, let's say we picked the following 20 sets of four-digit numbers from the random number table: 0132, 6783, 0856, 4353, 6904, 2569, 0237, 5903, 1230, 3451, 1101, 2470, 5076, 3151, 1732, 4827, 5732, 0634, 6079, 0293.

Convert the four-digit numbers to customer IDs

Using the selected four-digit numbers, convert them back to their respective customer IDs by removing any leading zeros where necessary. Using our example from Step 3, the customer IDs for our sample would be: 132, 6783, 856, 4353, 6904, 2569, 237, 5903, 1230, 3451, 1101, 2470, 5076, 3151, 1732, 4827, 5732, 634, 6079, 293.

Verify that the sample is random

To ensure that your sample of 20 customer IDs is random, check that the sample does not have any discernable patterns or repetitions. A random sample will not have customer IDs that follow a specific pattern or sequence. In our example, the 20 customer IDs do not appear to have any discernable patterns, so we have successfully selected a random sample of customers for the survey.

Key concepts

Random Number Table

A random number table is an important tool used in statistics to ensure fairness and prevent bias during random sampling processes. It's essentially a list of random digits arranged in rows and columns. These tables help in selecting sample numbers or elements without any predictable pattern.

When you use a random number table, every number in it has an equal chance of being chosen, making this method very reliable when aiming to achieve a truly random sample. For instance, in selecting customers for a survey, the access to this table allows you to pick customer IDs impartially. The randomness of selection eliminates potential biases that could influence the results of a survey. This is crucial when seeking to gain genuine insights from customer feedback.

• The randomness is crucial for removing bias.
• The table provides a systematic way of picking numbers.
• Each number in the table has an equal probability of selection.

Customer Survey

Conducting a customer survey is a valuable strategy for businesses to gather insights about their customers’ opinions, preferences, and satisfaction levels. This process involves directly reaching out to a sample of customers and asking them a series of questions designed to understand their experiences and expectations.

Customer surveys offer numerous benefits:

• They provide direct feedback from the people using their services or products.
• Surveys can uncover areas requiring improvement.
• They assist in developing strategies to enhance customer satisfaction.
• Data collected from surveys can guide in making informed business decisions.

By effectively utilizing the feedback from these surveys, businesses can make meaningful changes that translate into better services or products. The randomness of the sample ensures a representative group, revealing genuine insights rather than skewed data from non-representative feedback.

Random Sample Selection

Random sample selection is a technique in statistics where a subset of individuals is chosen from a larger set in such a way that each individual has an equal chance of being chosen. This concept is vital because it ensures the sample represents the population closely

There are several key points to understand about random sampling:

• This process guards against biases that could skew the results.
• It enhances the reliability and validity of survey results.
• By using a random method, the probability of every sample being chosen is equal, leading to more trustworthy outcomes.

In our exercise example, random sampling is used to choose 20 customer IDs from a list of 7000 using a random number table. This ensures that every customer within the population has the same opportunity to be part of the survey, fostering trust and credibility in the results obtained.

Four-Digit Numbers

Four-digit numbers are essential when working with a larger dataset, such as thousands of customer IDs. In instances where there are numerous items to choose from, such as 7000 customers, each item needs a unique identifier to ensure precise tracking and selection.

In the example we looked at, these identifiers range between 0001 and 7000. Each ID is four digits to guarantee consistent formatting and straightforward identification.

• A consistent four-digit format simplifies the selection process.
• Leading zeros are added for numbers less than 1000, which maintains uniformity.
• This ensures no overlap or confusion during sample selection.

By maintaining this four-digit scheme, we eliminate misinterpretations during data collection and further analytics, ensuring every chosen number unmistakably corresponds to only one unique customer in the list.