Draft A Response To Each Of The Bulleted Questions Be 954981

Draft A Response To Each Of The Bulleted Questions Below Each Questio

1. How would you describe "probability" from a Statistics for Business perspective to a person that has limited exposure to statistics?

From a business statistics perspective, probability quantifies the likelihood that a specific event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. For example, a 0.2 probability suggests a 20% chance that the event happens, helping businesses make informed decisions based on risk assessment and expected outcomes. Probability enables managers to evaluate uncertainties and plan strategies accordingly, making it a fundamental concept in analyzing patterns and forecasting future events.

2. Suppose you are rolling 2 dice. What would be the probability of getting 7?

The probability of rolling a sum of 7 with two dice is calculated by dividing the number of successful outcomes by the total possible outcomes. There are 6 combinations (1-6, 2-5, 3-4, 4-3, 5-2, 6-1) that sum to 7. Since each die has 6 sides, there are 36 possible outcomes in total. Therefore, the probability is 6 divided by 36, which simplifies to 1/6 or approximately 16.67%. This probability helps in understanding likelihoods in games of chance or risk modeling in business scenarios.

3. An airline would like to know the probability of a piece of passenger’s luggage weighing more than 40 pounds. To learn this probability, a baggage handler picks off every 20th bag from the conveyor and weighs it. Do you think these data allow the airline to use the Law of Large Numbers eventually to learn P (luggage weighs more than 40 pounds)?

The data collection method—sampling every 20th bag—can provide insights into the luggage weight distribution over time. Assuming the sampling is random and representative, and enough bags are weighed, the Law of Large Numbers suggests that the sample average will approach the true probability (P) as the sample size grows. Therefore, with a sufficiently large number of sampled bags, the airline can reliably estimate the probability that luggage exceeds 40 pounds. However, initial sample sizes must be large enough to ensure statistical reliability for the Law of Large Numbers to hold.

4. A basketball team is down by 2 points with only a few seconds remaining in the game. There’s a 50% chance that the team will be able to make a 2-point shot and tie the game, compared to a 30% chance that it will make a 3-point shot and win. If the game ends in a tie, the game continues to overtime. In overtime, the team has a 50% chance of winning. What should the coach do, go for the 2-point shot or the 3-point shot? Be sure to identify any assumptions you make.

The decision hinges on which shot maximizes the expected probability of winning. If the team opts for a 2-point shot: they have a 50% chance to tie, leading to overtime where they have a 50% chance of winning, resulting in an overall win probability of 0.5 * 0.5 = 25%. If they choose a 3-point shot: they have a 30% chance to win immediately, and a 70% chance the attempt fails, leaving the score as is and the game continuing. Assuming the team’s chances in overtime are independent of the shot type and that they can execute either shot effectively, taking the 3-point shot provides a higher immediate victory likelihood (30%) compared to a potential higher tie chance that leads to uncertain overtime outcomes. Therefore, the coach should opt for the 3-point shot, assuming the players are equally skilled at both shot types and that the higher immediate benefit outweighs the risk of losing the game outright.

5. A pharmaceutical company has developed a diagnostic test for a rare disease. The test has sensitivity 0.99 (the probability of testing positive among people with the disease) and specificity 0.995 (the probability of testing negative among people who do not have the disease). What other probability must the company determine in order to find the probability that a person who tests positive is in fact healthy?

The company needs to determine the prevalence of the disease in the population or the prior probability that a person has the disease. This baseline probability, combined with the sensitivity and specificity, is essential for applying Bayes’ theorem to calculate the positive predictive value—that is, the probability that someone who tests positive truly has the disease. Without knowing how common the disease is, the company cannot accurately determine the likelihood that a positive test indicates health or disease, which is critical in evaluating the test’s real-world effectiveness.

6. Some electronic devices are better used than new: The failure rate is higher when they are new than when they are six months old. For example, half of the personal music players of a particular brand have a flaw. If the player has the flaw, it dies in the first six months. If it does not have this flaw, then only 10% fail in the first six months. Yours died after you had it for three months. What are the chances that it has this flaw?

This problem involves conditional probability, which can be addressed using Bayes' theorem. Let F denote the flaw, and D denote failure within the first six months. Given that half of all devices have the flaw and 10% without flaws fail early, the probability that a device has the flaw, given that it failed, is calculated as:

P(F | D) = (P(D | F) * P(F)) / P(D)

where P(D) is the total probability of failure, computed as:

P(D) = P(D | F) P(F) + P(D | no F) P(no F)

Substituting known values: P(D | F) = 1, P(F)=0.5, P(D | no F)=0.1, P(no F)=0.5. This yields:

P(D) = (1)(0.5) + (0.1)(0.5) = 0.5 + 0.05 = 0.55.

Thus, the probability that the device has the flaw given that it failed is:

P(F | D) = (1 * 0.5) / 0.55 ≈ 0.91 or 91%. Therefore, it is highly likely that the device has the flaw, given it failed within three months.

7. A contractor built 30 similar homes in a suburban development. The homes have comparable size and amenities, but each has been sold with features that customize the appearance, landscape, and interior. The contractor expects the homes to sell for about $450,000. He expects that one-third of the homes will sell either for less than $400,000 or more than $500,000. Would a normal model be appropriate to describe the distribution of sale prices? What data would help you decide if a normal model is appropriate? (For example, the model is to describe the prices of as-yet-unsold homes.)

Given that one-third of the homes are anticipated to sell below $400,000 or above $500,000, the distribution appears to have heavier tails than a normal distribution, which suggests potential skewness or kurtosis issues. To assess whether a normal model is appropriate, data on historical prices, including measures of skewness and kurtosis, and the shape of the distribution (e.g., histograms or Q-Q plots), would be necessary. The mean and standard deviation of past sales, along with these shape characteristics, can help determine if the normal distribution fits well. A normal model with a mean around $450,000 and appropriate standard deviation could be consistent if the sale prices are symmetrically distributed with relatively few extreme outliers, but this needs confirmation from actual data.

Would a normal model be useful to describe the total size of adjustments when a CPA reviews tax forms? (For example, suppose the preliminary tax form claims that the business owes taxes of $40,000. The CPA then adjusts this amount.) What data would help decide if a normal model is appropriate? (You cannot use data from the current year.)

Using a normal model for tax adjustments can be appropriate if the adjustments tend to center around a mean value with symmetric variation. To evaluate this, historical data on past adjustments—such as average adjustment size, variability, skewness, and whether large adjustments are common—would be needed. Ideally, aggregate adjustment data over several years, segregated by business type or industry, would help determine the distribution shape. If the adjustments show a roughly symmetric distribution with no excessive outliers, a normal model can be suitable.

If the average adjustment for a particular CPA is -$7,000, then the standard deviation provides insight into the variability. For example, if σ is relatively small, then most adjustments are close to –$7,000; if large deviations are rare, then the normal model suggests that most businesses will experience similar adjustments. Finally, the fit of the normal model at the total level would depend on the consistency of adjustment data across all CPAs at the firm, requiring examining multidisciplinary historical data to validate the Gaussian assumption.

References

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