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The frequency distribution shows the number of medical tests conducted on 30 randomly selected emergency room patients. $$ \begin{array}{cc} \begin{array}{c} \text { Number of tests } \\ \text { performed } \end{array} & \begin{array}{c} \text { Number of } \\ \text { patients } \end{array} \\ \hline 0 & 11 \\ 1 & 9 \\ 2 & 5 \\ 3 & 4 \\ 4 \text { or more } & 1 \end{array} $$ If a patient is selected at random, find these probabilities: a. The patient had exactly 3 tests done. b. The patient had at most 2 tests done. c. The patient has 1 or 2 tests done. \(d\). The patient had fewer than 3 tests done. e. The patient had at least 3 tests done.

Step by step solution

01

Determine the Total Number of Patients

The total number of patients is the sum of patients in each category: \[11 + 9 + 5 + 4 + 1 = 30\]Thus, there are 30 patients in total.

02

Determine Probability for Exact Tests - 3 Tests

To find the probability that a patient had exactly 3 tests done, take the number of patients with 3 tests and divide it by the total number of patients:\[P( ext{3 tests}) = \frac{4}{30}\]This probability simplifies to \(\frac{2}{15}\).

03

Calculate Probability for At Most 2 Tests

To determine the probability that a patient had at most 2 tests, sum the number of patients with 0, 1, or 2 tests and divide by the total:\[P( ext{at most 2 tests}) = \frac{11 + 9 + 5}{30} = \frac{25}{30}\]Simplifying this gives \(\frac{5}{6}\).

04

Find Probability for 1 or 2 Tests

Add the number of patients with 1 or 2 tests, then divide by the total number of patients:\[P( ext{1 or 2 tests}) = \frac{9 + 5}{30} = \frac{14}{30}\]This reduces to \(\frac{7}{15}\).

05

Calculate Probability for Fewer Than 3 Tests

Add the number of patients with fewer than 3 tests (i.e., 0, 1, or 2) and divide by the total number:\[P( ext{fewer than 3 tests}) = \frac{11 + 9 + 5}{30} = \frac{25}{30}\]This simplifies to \(\frac{5}{6}\).

06

Calculate Probability for At Least 3 Tests

Sum the number of patients with 3 or more tests and divide by the total:\[P( ext{at least 3 tests}) = \frac{4 + 1}{30} = \frac{5}{30}\]Simplify this fraction to get \(\frac{1}{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Distribution

When we talk about frequency distribution in statistics, we're referring to a way of organizing data that shows the frequency, or the number of times, each value appears. In the context of our exercise, the frequency distribution is being used to show the number of medical tests conducted on a sample of 30 emergency room patients.

Here's how it works:

• Value: The number of tests each patient undergoes—represented as 0, 1, 2, 3, or 4 or more.
• Frequency: How many patients received that specific number of tests.

For example, 11 patients had no tests, 9 had one test, and so on. By organizing data this way, it becomes easier to analyze and understand the probability of different outcomes, like whether a randomly chosen patient had a particular number of tests.

Random Selection

Random selection is a crucial principle in probability and statistics. This method ensures that every individual has an equal chance of being chosen. In our exercise, this allows us to analyze the patients' data objectively.

Why is random selection important?

• It eliminates bias, ensuring that the data reflects the true characteristics of the population.
• It provides a fair representation of the group, allowing us to make accurate probability calculations.

In practical terms for our problem, it means the choice of any patient from the total of 30 is unbiased. Thus, all probability calculations are valid and truly represent the scenario being analyzed.

Emergency Room Patients

Understanding the context of the problem is vital. Here, we're looking at data from emergency room patients. This specific setting often implies a wide variety of medical needs, hence the possible range of tests from 0 to 4 or more.

Each patient entering the emergency room may have differing needs, so the number of tests conducted can vary greatly:

• Some might require no diagnostic tests due to obvious ailments.
• Others might need multiple tests to diagnose the issue correctly or rule out possibilities.

The frequency distribution data we have acts as a real-life snapshot, helping us understand how resources like tests are allocated in emergency situations.

Step by Step Solution

Breaking down a complicated problem into simpler, smaller steps makes it manageable. Let's review each step used in solving the problem:

• Determine Total Patients: First, all patient records are summed up to understand the whole data range, which here is 30 patients.
• Calculate Specific Probabilities: To find out the probability of discrete outcomes (like a patient having 3 tests), divide the number of occurrences by total patients.
• Simplify Fractions: Probabilities are often given as fractions, which are simplified to offer easier understanding and comparison.
• Consider Broader Cases: We evaluate wider conditions such as 'at most' and 'at least,' grouping relevant frequencies to find their probability.

These steps guide the problem-solving process, making complex probability queries simple and systematic.