Question

A population consists of \(N=500\) experimental units. Use a random number table to select a random sample of \(n=20\) experimental units. (HINT: Since you need to use three-digit numbers, you can assign 2 three-digit numbers to each of the sampling units in the manner shown in the table.) What is the probability that each experimental unit is selected for inclusion in the sample?

Short answer

Answer: The probability that each experimental unit is selected for inclusion in the random sample is \(\frac{1}{500}\).

Step-by-step solution

Assign three-digit numbers to each experimental unit

First, we will assign two three-digit numbers to each experimental unit in the population of N=500. There will be a total of 1000 three-digit numbers.

Use a random number table to select 20 experimental units

Now, use a random number table to select 20 three-digit numbers. If any of the chosen three-digit numbers match with the assigned numbers of the experimental units, those experimental units will be included in the random sample.

Calculate the probability of each experimental unit being selected

To calculate the probability that each experimental unit is selected for inclusion in the sample, we must first determine the total number of ways to select 20 experimental units from 500. Since each experimental unit is assigned 2 unique three-digit numbers, there are 1000 three-digit numbers available. We are selecting 20 three-digit numbers from the 1000 available numbers. Therefore, the probability that a specific experimental unit is selected is: \(\frac{\text{Number of three-digit numbers assigned to experimental unit}}{\text{Total three-digit numbers available}} \times \frac{\text{Number of three-digit numbers we want to select}}{\text{Total three-digit numbers we can select from}}\) Which simplifies to: \(\frac{2}{1000} \times \frac{20}{20}\)

Simplify the probability

We can now simplify the expression found in step 3 to find the probability of each experimental unit being selected: \(\frac{2}{1000} \times \frac{20}{20} = \frac{2}{1000} = \frac{1}{500}\) So the probability that each experimental unit is selected for inclusion in the sample is \(\frac{1}{500}\).

Key concepts

Probability Calculation

When it comes to probability calculations in statistics, we often deal with the likelihood of a specific event occurring from a range of possible outcomes. In this particular case, we're interested in the probability that each experimental unit, out of a total of 500, is selected for inclusion in a sample.

To calculate this, we need to understand that each experimental unit is associated with two separate three-digit numbers, giving a total of 1000 numbers.
Using a random selection method like a random number table means choosing 20 numbers out of these 1000.

The probability for one specific experimental unit to be selected is calculated by the formula:

• Determine how many ways the event can occur. Here, it's important to realize that each unit has 2 chances in a pool of 1000 numbers.
• Since 20 numbers are selected, each unit has a chance of being selected through one of its numbers, computed as \( \frac{2}{1000} \times \frac{20}{20} \).
• When simplified, the probability boils down to \( \frac{1}{500} \), meaning each experimental unit has a one in 500 chance of being included in the sample.

Random Number Table

A random number table is a tool that's commonly used in statistical sampling methods. It consists of a sequence of random digits arranged in tabular form. Researchers use it to ensure randomness when selecting a sample from a larger population, preventing bias and ensuring each member of the population has an equal chance of selection.

In the task described, the random number table helps select 20 experimental units from a population of 500. Because each experimental unit is assigned two unique three-digit numbers, the table randomizes which units get picked by ensuring selections aren't influenced by the researcher's expectations or biases.

To use a random number table effectively:

• Assign unique identifiers, like three-digit numbers, to each member of the population.
• Read the table according to the specified number of digits to find your random sample.
• Assign these numbers to ensure each experimental unit is given equal opportunity to be selected.

Experimental Units

Experimental units are the basic objects of study to which treatments are applied and upon which data is collected. These can vary widely in research: from individual humans to plants, animals, plots of land, or even objects like parts or components in manufacturing studies.

In our context, we have 500 experimental units, each given equal importance in the study. Their identification through unique three-digit numbers ensures that each can equally contribute to the random sampling process aimed at drawing a representative sample.

Key things to understand about experimental units:

• They must be clearly defined and distinguishable from one another in research.
• Assignment of unique identifiers, like numbers, prevents confusion and ensures every unit is distinct.
• Involvement of all units in the random number selection process ensures valid statistical estimates, avoiding biased results.