Problem 13 - Basic Statistics for Business and Economics with Student

Question

An American Society of Investors survey found 30 percent of individual investors use a discount broker. In a random sample of nine individuals, what is the probability: a. Exactly two of the sampled individuals have used a discount broker? b. Exactly four of them have used a discount broker? c. None of them have used a discount broker?

Step-by-Step Solution

Answer

a. 0.199, b. 0.122, c. 0.040

Step 1: Understanding the Problem

This problem involves finding the probability of a specific number of successes in a fixed number of trials. This is known as a binomial probability problem where we have a probability of success (using a discount broker) of 30% or 0.3, and we want to know the probabilities for certain numbers of individuals out of a sample of 9.

Step 1: Define Binomial Probability Formula

The binomial probability formula is given by\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success.

Step 2: Calculate Probability for Exactly Two

For exactly two individuals using a discount broker, we set \(n = 9\), \(k = 2\), and \(p = 0.3\). The probability is then\[P(X = 2) = \binom{9}{2} (0.3)^2 (0.7)^{7}\].Calculating:\[\binom{9}{2} = 36\]\[(0.3)^2 = 0.09\]\[(0.7)^7 \approx 0.0823543\].Thus, \[P(X = 2) = 36 \times 0.09 \times 0.0823543 \approx 0.199\].

Step 3: Calculate Probability for Exactly Four

For exactly four individuals using a discount broker, we set \(n = 9\), \(k = 4\), and \(p = 0.3\). The probability is then\[P(X = 4) = \binom{9}{4} (0.3)^4 (0.7)^{5}\].Calculating:\[\binom{9}{4} = 126\]\[(0.3)^4 = 0.0081\]\[(0.7)^5 \approx 0.16807\].Thus, \[P(X = 4) = 126 \times 0.0081 \times 0.16807 \approx 0.122\].

Step 4: Calculate Probability for None

For none of the individuals using a discount broker, we set \(n = 9\), \(k = 0\), and \(p = 0.3\). The probability is then\[P(X = 0) = \binom{9}{0} (0.3)^0 (0.7)^{9}\].Calculating this gives:\[\binom{9}{0} = 1\]\[(0.3)^0 = 1\]\[(0.7)^9 \approx 0.0403536\].Thus, \[P(X = 0) = 1 \times 1 \times 0.0403536 = 0.040\].

Key Concepts

Probability Theory

Probability theory is an essential branch of mathematics that deals with the likelihood of events occurring. In real-life scenarios, it helps us predict the probability of different outcomes. When dealing with binomial probabilities, the situation involves a fixed number of trials and a constant probability of success for each trial.

• A **binomial probability problem** requires you to determine the chance of a specific number of "successes" (like using a discount broker) out of a certain "n" number of trials (such as the number of individuals surveyed).
• The formula \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\) explains how to calculate this probability. This involves choosing "k" successes out of "n" trials, where "p" is the probability of a single success, while \(1-p\) is the probability of a failure.
• Each scenario—whether none, two, or four people using a broker—requires substituting these values into the formula.

Statistics for Business

Statistics for business uses tools like probability theory to make informed decisions based on data. Understanding probability allows businesses to assess risks and outcomes, potentially leading to smarter investment strategies.

• **Risk assessment**: Businesses can predict the chance of achieving goals such as a certain level of investment success by estimating probabilities of specific outcomes.
• This exercise is an example of how to determine the probability of customers choosing a specific service, which helps in marketing and strategic planning.
• The calculation of probabilities in these problems can guide business decisions, allowing firms to allocate resources more strategically.

Mathematical Statistics

Mathematical statistics involves applying theoretical mathematics to understand, interpret, and manipulate data. Managing concepts like mean, variance, and probabilities lies at its core.

• A binomial distribution, which models situations with two potential outcomes (such as success or failure), is a fundamental part of mathematical statistics.
• By calculating probabilities using tools such as the binomial formula, we can interpret data more effectively. For instance, knowing the probability of a certain number of individuals choosing a service informs understanding of broader trends.
• Since mathematical statistics is deeply integrated into various scientific inquiries and business analyses, mastering these calculations is key for data-driven decision making.