Question

A study is to be conducted in a hospital to determine the attitudes of nurses toward various administrative procedures. If a sample of 10 nurses is to be selected from a total of 90 , how many different samples can be selected? (HINT: Is order important in determining the makeup of the sample to be selected for the survey?)

Short answer

Answer: 43,758,513 different samples can be selected.

Step-by-step solution

Determine values for n and k

In this problem, n represents the total number of nurses (90), and k represents the number of nurses to be selected for the sample (10).

Apply the Combination Formula

Using the values for n and k determined in Step 1, the combination formula can be applied to find the number of possible samples: C(90, 10) = 90! / (10! (90-10)!)

Calculate Factorials

Calculate the factorial for each value in the formula: 90! = 90 * 89 * ... * 3 * 2 * 1 10! = 10 * 9 * 8 * ... * 3 * 2 * 1 (90-10)! = 80! = 80 * 79 * ... * 3 * 2 * 1

Substitute Factorials Back into Formula

Replace the factorials in the combination formula with the calculated values: C(90, 10) = (90!)/(10! * 80!)

Simplify and Calculate the Number of Samples

Simplify the combination formula and solve for the number of possible samples: C(90, 10) = (90 * 89 * ... * 81)/(10 * 9 * 8 * ... * 2 * 1) C(90, 10) = 43,758,513 So, there are 43,758,513 different samples that can be selected from a total of 90 nurses.

Key concepts

Combination Formula

The combination formula is a central concept in combinatorics. It is used when the order in which items are selected does not matter. In our exercise, we are selecting a sample of nurses from a larger group without regard to the sequence in which they are selected. This makes it a perfect scenario for using combinations.

The formula for combinations is:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

Where:

• \( n \) is the total number of items to choose from.
• \( k \) is the number of items to be chosen.
• \( ! \) denotes a factorial, which we will explain in more detail below.

Using the combination formula, you can easily calculate the number of ways to select a subset of items from a larger set. This tool is vital when dealing with questions that do not depend on the sequence, like selecting a jury or drawing cards from a deck.

Factorials

Factorials play a crucial role in computing combinations. In essence, a factorial is the product of all positive integers up to a specified number, represented by the exclamation point symbol (!). Understanding factorials is essential for applying the combination formula correctly.

Let's break this down with examples:

• \( n! = n \times (n-1) \times (n-2) \cdots 3 \times 2 \times 1 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

In our nurse selection problem, the factorials are used to calculate both the numerator and the denominator of the combination formula. Calculating larger factorials is straightforward yet requires care to ensure accuracy, especially as numbers grow exponentially large, like \( 90! \) in our example. The factorial operation simplifies many complex problems, rendering them manageable and solvable.

Sample Selection

Sample selection is the process of choosing individuals from a larger population to participate in a study or survey. It is a fundamental aspect of statistical research, as it affects the reliability and generalizability of the results.

In the given problem, we're dealing with selecting 10 nurses from a total of 90. The exercise specifies that order does not matter, aligning with the criteria for using combinations. Choosing a sample wisely ensures the group represents the larger population well. This enhances the accuracy of conclusions drawn from the study.

Key points:

• Order is irrelevant, so combinations are appropriate.
• Sample size (10) is significantly smaller than the population (90), a common scenario in sampling.
• The selection process should aim to minimize bias and increase representativeness of the overall population.

Understanding sample selection and its implications help in making informed decisions about the ways results can be utilized practically.

Statistical Methods

Statistical methods involve a suite of techniques used to collect, analyze, interpret, and present data. In our context, they help to make informed predictions and decisions based on data. Knowing which statistical method to use is crucial, as it impacts the research's findings.

When you calculate combinations, like \( C(90, 10) \), you are applying a statistical method to explore possible outcomes. This method falls under probability and statistics, analyzing situations where items are grouped without considering order.

Statistical methods often involve:

• Descriptive statistics, for summarizing data.
• Inferential statistics, for deriving conclusions beyond the immediate data.
• Probability, which helps understand and interpret the likelihood of different outcomes, such as selecting sample groups.

Leveraging statistical methods, you enhance the precision of predictions and results, allowing for conclusions that can guide decision-making processes and future research.