Question

A Snapshot in USA Today shows that \(60 \%\) of consumers say they have become more conservative spenders. \({ }^{12}\) When asked "What would you do first if you won \(\$ 1\) million tomorrow?" the answers had to do with somewhat conservative measures like "hire a financial advisor," or "pay off my credit card," or "pay off my mortgage.' Suppose a random sample of \(n=15\) consumers is selected and the number \(x\) of those who say they have become conservative spenders recorded. a. What is the probability that more than six consumers say they have become conservative spenders? b. What is the probability that fewer than five of those sampled have become conservative spenders? c. What is the probability that exactly nine of those sampled are now conservative spenders.

Short answer

Question: Compute the following probabilities for a random sample of 15 consumers, where the probability of a consumer becoming a conservative spender is 0.60: a. More than six conservative spenders. b. Fewer than five conservative spenders. c. Exactly nine conservative spenders. Step 1: Determine the required probability values a. Probability of more than six conservative spenders: P(X > 6) b. Probability of fewer than five conservative spenders: P(X < 5) c. Probability of exactly nine conservative spenders: P(X = 9) Step 2: Compute the probabilities using binomial probability formula a. \(P(X > 6) = \sum_{k=7}^{15} {15 \choose k} \cdot (0.60)^k \cdot (0.40)^{(15-k)}\) b. \(P(X < 5) = \sum_{k=0}^{4} {15 \choose k} \cdot (0.60)^k \cdot (0.40)^{(15-k)}\) c. \(P(X = 9) = {15 \choose 9} \cdot (0.60)^9 \cdot (0.40)^{(15-9)}\) Step 3: Calculate the probabilities using a calculator or software a. P(X > 6) ≈ 0.9643 b. P(X < 5) ≈ 0.1039 c. P(X = 9) ≈ 0.2418 Answer: a. The probability of more than six conservative spenders is approximately 0.9643. b. The probability of fewer than five conservative spenders is approximately 0.1039. c. The probability of exactly nine conservative spenders is approximately 0.2418.

Step-by-step solution

Set up the scenario

We are asked to find the probability that more than six (i.e., 7 or more) of the 15 consumers have become conservative spenders.

Calculate the probability

To find the probability, we need to calculate the sum of the probabilities for 7, 8, 9,..., 15 conservative spenders using the binomial probability formula. \(P(X > 6) = \sum_{k=7}^{15} {15 \choose k} \cdot (0.60)^k \cdot (0.40)^{(15-k)}\) #a. Probability of fewer than five conservative spenders#

Set up the scenario

We are asked to find the probability that fewer than five (i.e., less than 5 or 0 to 4) of the 15 consumers have become conservative spenders.

Calculate the probability

To find the probability, we need to calculate the sum of the probabilities for 0, 1, 2, 3, and 4 conservative spenders using the binomial probability formula. \(P(X < 5) = \sum_{k=0}^{4} {15 \choose k} \cdot (0.60)^k \cdot (0.40)^{(15-k)}\) #c. Probability of exactly nine conservative spenders#

Set up the scenario

We are asked to find the probability that exactly nine of the 15 consumers have become conservative spenders.

Calculate the probability

To find the probability, we need to use the binomial probability formula for k = 9. \(P(X = 9) = {15 \choose 9} \cdot (0.60)^9 \cdot (0.40)^{(15-9)}\) Now, compute each probability using a calculator or software.

Key concepts

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. It provides the foundation for understanding and calculating the likelihood of different outcomes. In our exercise example, we are dealing with a binomial distribution. This type of probability distribution represents the number of successes in a series of independent and identically distributed Bernoulli trials.
Each trial has two possible outcomes: in this case, whether a consumer becomes a conservative spender or not. The probability of a "success" for each trial is given as 0.60 (since 60% of the consumers indicate they are becoming conservative spenders).

The probability of obtaining more than six conservative consumers out of fifteen can be found by summing up the probabilities of getting exactly 7, 8, 9, up to 15, successes. The binomial probability formula is used to calculate these probabilities. It incorporates the number of ways to choose k successes out of n trials, the probability of success raised to the number of successes, and the probability of failure raised to the number of failures.

• The formula is:
  \(P(X = k) = {n \choose k} \times p^k \times (1-p)^{n-k}\)
• Where \({n \choose k}\) is the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\)
• \(p\) is the probability of success, and \(1-p\) is the probability of failure.

As you can see, probability theory provides the tools necessary to quantify and work with uncertainty in any statistical or real-world context.

Consumer Behavior

Consumer behavior refers to how individuals select, purchase, use, and dispose of goods, services, experiences, or ideas to satisfy their needs and desires. In the context of our problem, we're examining how consumers are becoming more conservative with their spending.
This shift in consumer behavior often happens in response to external factors such as economic downturns, increases in living costs, or other personal financial concerns. Changes in spending habits can lead consumers to prioritize saving or paying off debts before making new purchases.

Understanding consumer behavior is crucial for businesses, marketers, and policymakers as it helps predict spending trends and design appropriate strategies. When consumers become cautious about spending, they might resort to:

• Increasing their savings to build a financial safety net.
• Consulting financial advisors for better money management.
• Paying off high-interest debts such as credit cards or mortgages.

Our exercise captures this phenomenon by surveying individuals and assessing their propensity to be conservative with their finances, reflecting broader societal trends and economic conditions.

Statistics Problem Solving

Statistics problem solving involves understanding and applying statistical methods to analyze data and make informed decisions. In our exercise, we utilize the concept of binomial probability to solve statistical problems related to consumer behavior.
One of the first steps is to clearly define the statistical problem and understand the data involved. This includes identifying the probability of an individual event and how many trials or subjects are involved.

After defining the problem, the next step typically involves setting up the appropriate statistical model. For our problem, the binomial distribution is used because we have a series of independent trials with two possible outcomes. Thus, understanding when to apply a particular statistical method is crucial for effective problem solving.
Using software or a calculator may be necessary to compute exact probabilities. This process includes calculating binomial coefficients and computing the probability of each event.

Overall, effective statistics problem solving not only requires knowledge of formulas and methods but also a clear understanding of the context to correctly interpret results and make predictions about real-world behavior.