Problem 17 - Basic Statistics for Business & Economics

Question

A recent survey by the American Accounting Association revealed \(23 \%\) of students graduating with a major in accounting select public accounting. Suppose we select a sample of 15 recent graduates. a. What is the probability two select public accounting? b. What is the probability five select public accounting? c. How many graduates would you expect to select public accounting?

Step-by-Step Solution

Answer

a) 0.231, b) 0.237, c) 3.45 graduates.

Step 1: Understanding the Problem

We need to find the probability that a certain number of accounting graduates choose public accounting from a sample of 15. Additionally, we need to calculate the expected number of graduates selecting public accounting. Since this is a binomial probability problem, where each graduate can either select public accounting or not, we will use the binomial probability formula.

Step 2: Defining the Parameters

Define the parameters for a binomial distribution: number of trials \( n = 15 \), probability of success (choosing public accounting) \( p = 0.23 \). We will calculate the probability of exactly 2 and 5 successes and also determine the expected number of graduates choosing public accounting.

Step 3: Using the Binomial Probability Formula

The binomial probability formula is given by \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( \binom{n}{k} \) is the binomial coefficient.

Step 4: Calculating Probability for Two Graduates

Using the formula for \( k = 2 \): \[ P(X = 2) = \binom{15}{2} (0.23)^2 (1-0.23)^{15-2} \]Calculate \( \binom{15}{2} = \frac{15 \times 14}{2 \times 1} = 105 \).Then, \( (0.23)^2 = 0.0529 \) and \( (0.77)^{13} \approx 0.047893 \).So, \[ P(X = 2) \approx 105 \times 0.0529 \times 0.047893 \approx 0.231 \]

Step 5: Calculating Probability for Five Graduates

For \( k = 5 \):\[ P(X = 5) = \binom{15}{5} (0.23)^5 (1-0.23)^{15-5} \]Calculate \( \binom{15}{5} = 3003 \).Then, \( (0.23)^5 \approx 0.002985983 \) and \( (0.77)^{10} \approx 0.196874404 \).So, \[ P(X = 5) \approx 3003 \times 0.002985983 \times 0.196874404 \approx 0.237 \]

Step 6: Finding the Expected Number of Graduates

The expected number of graduates selecting public accounting can be found using the formula for expected value of a binomial distribution, \( E(X) = n \times p \).Substitute the given values to find the expected number: \[ E(X) = 15 \times 0.23 = 3.45 \] graduates.

Key Concepts

Probability

Probability is the likelihood of an event happening. In this context, it refers to how likely it is that a certain number of accounting graduates choose public accounting. The probability value is always between 0 and 1, where 0 means an impossible event, and 1 means a certain event. The probability of different outcomes can be calculated with various mathematical tools, such as probability formulas and equations.
In binomial experiments, such as the one with the accounting graduates, each trial is independent which means the outcome of one does not affect the next. For each graduate, there are two possible outcomes: they either choose public accounting or they don't.

• When we calculate the probability of two graduates choosing public accounting, we are looking for a specific outcome from a larger sample, which involves mathematical calculations based on combinations and the probability of individual events.

Expected Value

The expected value is a predicted value of a variable and is often seen as the average outcome if the experiment were to be repeated many times. In a binomial distribution, like the graduates, the expected value helps to determine a central tendency.
The formula to find the expected value, which is denoted as \( E(X) \), is the product of the number of trials \( n \) and the probability of success \( p \). This formula is given by \( E(X) = n \times p \).

• This means for our graduates scenario: if 15 students are graduating, 23% are expected to go into public accounting, so we'd expect around 3 or 4 students (when rounded) to choose this path.
• This method gives an average expectation in the long run, helping to understand the overall behavior of the distribution.

Binomial Probability Formula

The binomial probability formula is essential for solving problems where there are two possible outcomes, like our graduates situation. The formula calculates the probability that a given number of successes, that is, students choosing public accounting, will occur out of a fixed number of trials.
The formula is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \).

• Here, \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials.
• \( p \) is the probability of success on an individual trial, and \( 1-p \) is the probability of failure.

Grab your calculator for these computations, as they involve multiplying large numbers.The binomial probability begins to give you a clearer picture of how likely certain outcomes are, helping with predictions based on initial parameters.
You can break down the calculation steps just like in the example to determine specific counts of success.