Problem 42 - Basic Statistics for Business and Economics with Student

Question

Tire and Auto Supply is considering a 2 -for- 1 stock split. Before the transaction is finalized, at least two thirds of the 1,200 company stockholders must approve the proposal. To evaluate the likelihood the proposal will be approved, the CFO selected a sample of 18 stockholders. He contacted each and found 14 approved of the proposed split. What is the likelihood of this event, assuming two-thirds of the stockholders approve?

Step-by-Step Solution

Answer

The probability is approximately 15.4%.

Step 1: Identify the Parameters

We need to determine the probability that 14 out of 18 stockholders approve if 2/3 of all stockholders are expected to approve. This is a binomial probability problem. The number of trials (n) is 18 and the probability of success (approval) on each trial (p) is 2/3.

Step 2: Use the Binomial Formula

The binomial probability formula is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( n = 18 \), \( k = 14 \), \( p = \frac{2}{3} \), and \( 1-p = \frac{1}{3} \).

Step 3: Calculate the Combination

\[ \binom{18}{14} = \frac{18!}{14!(18-14)!} = \frac{18 \times 17 \times 16 \times 15}{4 \times 3 \times 2 \times 1} = 3{,}060 \]

Step 4: Calculate Probability of Each Term

Calculate \( p^k = \left(\frac{2}{3}\right)^{14} \) and \( (1-p)^{n-k} = \left(\frac{1}{3}\right)^{4} \). These calculations give approximately \( 0.0041 \) and \( 0.0123 \) respectively.

Step 5: Combine Results

Plug these into the binomial formula: \[ P(X = 14) = 3{,}060 \times 0.0041 \times 0.0123 \approx 0.154 \]

Step 6: Interpretation

The probability of having exactly 14 approvals out of 18 is approximately 15.4%. This suggests there is a moderate likelihood of this event occurring.

Key Concepts

Binomial Distribution

A binomial distribution is a statistical method that models the number of successful outcomes in a specific number of trials. Each trial is independent, meaning the outcome of one does not affect the others, and there are only two possible outcomes: success or failure. In our example, the success is when a stockholder approves the proposal.
The key parameters for the binomial distribution include:

• Number of Trials (): This represents how many times the experiment is conducted. Here, n is 18 as the CFO surveys 18 stockholders.
• Probability of Success (p): This is the probability of approval by a stockholder. It is given as \( \frac{2}{3} \) in our scenario.
• Number of Successes (k): We seek the probability that 14 out of these 18 stockholders approve the proposal.

Understanding these elements is crucial for calculating probabilities using the binomial formula, as they define how the distribution is structured and applied to the problem.

Stockholders Approval

When a company considers significant changes, like a stock split, they often require stockholders' approval. This approval process embodies democratic decision-making in corporate governance, ensuring that the interests of the majority align with the company's direction.
In this specific problem, Tire and Auto Supply is considering a 2-for-1 stock split. To proceed, two-thirds of their 1,200 stockholders need to approve it. This shows the importance of having stockholders on board for major decisions.
Approval from more than two-thirds represents a substantial majority, highlighting the significant threshold needed for such corporate decisions. Finding how likely it is that a representative sample of stockholders agrees provides insight into the overall likelihood of approval from the entire group.

Probability Calculation

Probability calculation in this scenario involves applying the binomial probability formula to model the likelihood of a specific number of approved votes in the selected sample. The formula used is:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]This formula calculates the probability of exactly k successes (approvals) in n trials (stockholders contacted), each with a probability p of success.
Here's how it works step-by-step:

• Calculate the Combination: Find \( \binom{n}{k} \), the number of ways to choose k successes from n trials. For our problem, this was calculated as 3,060.
• Determine Probability for Each Term: Compute \( p^k \) and \( (1-p)^{n-k} \). Here, \( p^k \) was approximately 0.0041 and \( (1-p)^{n-k} \) was about 0.0123.
• Combine Results: Multiply these values together to get the final probability, \( P(X = 14) \), which comes to approximately 0.154 or 15.4%.

These calculations show that there is a moderate chance that exactly 14 out of the 18 stockholders approve the proposal.

Statistical Analysis

Statistical analysis involves using statistical methods to interpret data and draw conclusions from the data set. In this case, the analysis focused on evaluating the likelihood of stockholder approval through a sample, offering a predictive insight into the larger population.
By analyzing approval from a smaller group of 18 stockholders, the CFO uses probability to predict overall approval trends. This analysis helps in offering data-backed insights. If the sampled probability of approval is high, it suggests that the full stockholder group might also likely approve the proposal.
This kind of analysis is important because it can influence decision-making, risk assessment, and planning. It grounds corporate strategies in statistical evidence, potentially leading to more informed and strategic choices.