Problem 56 - Basic Statistics for Business & Economics

Question

According to sales information in the first quarter of \(2016,2.7 \%\) of new vehicles sold in the United States were hybrids. This is down from \(3.3 \%\) for the same period a year earlier. An analyst's review of the data indicates that the reasons for the sales decline include the low price of gasoline and the higher price of a hybrid compared to similar vehicles. Let's assume these statistics remain the same for 2017 . That is, \(2.7 \%\) of new car sales are hybrids in the first quarter of 2017 . For a sample of 40 vehicles sold in the Richmond, Virginia, area: a. How many vehicles would you expect to be hybrid? b. Use the Poisson distribution to find the probability that five of the sales were hybrid vehicles. c. Use the binomial distribution to find the probability that five of the sales were hybrid vehicles.

Step-by-Step Solution

Answer

a. 1.08 (expected hybrids); b. 0.0028 (Poisson); c. 0.0030 (Binomial).

Step 1: Determine Expected Number of Hybrid Vehicles (Part a)

To find the expected number of hybrid vehicles sold, multiply the total number of vehicles in the sample by the percentage of hybrid vehicles \(2.7\%\). \[ \text{Expected number of hybrids} = 40 \times \frac{2.7}{100} = 40 \times 0.027 = 1.08 \]So, we expect approximately 1.08 hybrids.

Step 2: Calculate Probability Using Poisson Distribution (Part b)

For the Poisson distribution, the expected number of hybrids \((\lambda)\) is 1.08. Use the Poisson probability formula:\[P(X=k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}\]where \(k = 5\).\[P(X=5) = \frac{e^{-1.08} \cdot (1.08)^5}{5!} \approx \frac{0.339595}{120} \approx 0.002831\]So, the probability of selling 5 hybrids is approximately 0.0028.

Step 3: Calculate Probability Using Binomial Distribution (Part c)

For the binomial distribution, the number of trials \((n)\) is 40, and the probability of success \((p)\) is 0.027. Use the binomial probability formula:\[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \(k = 5\).\[P(X=5) = \binom{40}{5} \times 0.027^5 \times (1-0.027)^{35} \approx 658008 \times 0.00000005485 \times 0.397555 \approx 0.002976\]Thus, the probability of selling 5 hybrids is approximately 0.0030.

Key Concepts

Binomial Distribution

The binomial distribution is a common method used to model the number of successful outcomes in a fixed number of trials. Each trial is independent and has only two possible outcomes: success or failure. For example, when mailing a letter, the two possible outcomes might be that the letter either arrives (success) or does not arrive (failure). In our problem, the number of trials is 40 (the sampled vehicles), and the probability of success is 2.7% (the chance that a vehicle is a hybrid). The formula used to find the probability in a binomial distribution is: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\] Where:

• \(n\) is the number of trials
• \(k\) is the number of successes (hybrids sold in this case)
• \(p\) is the probability of a success on each trial

In the context of the problem, we're finding the probability that exactly 5 out of 40 cars sold are hybrids. This involves plugging the appropriate values into our binomial formula to get an approximate probability of 0.0030. Remember, when using the binomial distribution, all trials are assumed to be independent, and the probability of success remains constant throughout.

Probability Calculation

Probability calculation is at the heart of statistics and involves determining how likely an event is to occur. It's expressed as a number between 0 and 1, where 0 means the event won't happen, and 1 means it definitely will. Calculating probabilities can involve different methods, depending on the type of distribution, such as binomial, Poisson, or normal distributions. In our exercise, we needed to calculate probabilities using both the Poisson and binomial distributions. In general:

• The Poisson distribution is typically used for modeling rare events over a fixed period or area and is characterized by the mean number of occurrences (\(\lambda\)).
• The binomial distribution is used when you want to know the probability of a certain number of successes in a specific number of trials.

Performing these calculations involves using specific formulas for each type of distribution. For instance, given the Poisson distribution formula \(P(X=k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!}\), you can find the probability of exactly \(k\) events (hybrids sold) happening. Probability calculations help us understand how likely an event is and can be used in decision-making and predicting outcomes.

Expected Value

The expected value is essentially the average or mean expected outcome of a random variable over many trials. It's a crucial concept in probability, often used to determine what to anticipate in a probabilistic event. The expected value is calculated as: \[\text{Expected value} = n \times p\] Where:

• \(n\) is the total number of trials or observations
• \(p\) is the probability of success on a single trial

For the problem at hand, we calculated the expected number of hybrid vehicles sold by multiplying the sample size (40 vehicles) by the probability of selling a hybrid (2.7%). Therefore, the expected value gives us 1.08, suggesting that, on average, just over one hybrid would be sold out of the sample of 40 vehicles. Expected value provides a long-term average and is extremely useful in fields like economics, insurance, and any scenario involving risk and uncertainty. However, note that the actual outcome might differ from the expected value in any given situation, especially with a small sample size.