91Ó°ÊÓ

Suppose you're considering buying your expensive chemistry textbook on Ebay. Looking at past auctions suggests that the prices of this textbook follow an approximately normal distribution with mean $$\$ 89$$ and standard deviation $$\$ 15$$. (a) What is the probability that a randomly selected auction for this book closes at more than $$\$ 100 ?$$ (b) Ebay allows you to set your maximum bid price so that if someone outbids you on an auction you can automatically outbid them, up to the maximum bid price you set. If you are only bidding on one auction, what are the advantages and disadvantages of setting a bid price too high or too low? What if you are bidding on multiple auctions? (c) If you watched 10 auctions, roughly what percentile might you use for a maximum bid cutoff to be somewhat sure that you will win one of these ten auctions? Is it possible to find a cutoff point that will ensure that you win an auction? (d) If you are willing to track up to ten auctions closely, about what price might you use as your maximum bid price if you want to be somewhat sure that you will buy one of these ten books?

Step by step solution

01

Understand the Problem for Part (a)

We need to find the probability that a randomly selected auction closes at more than $100. We will use the properties of the normal distribution with a mean (\( \mu \)) of 89 and a standard deviation (\( \sigma \)) of 15. The probability we are looking for is\( P(X > 100) \).

02

Convert to a Standard Normal Variable

Convert the auction price to a standard normal variable using the formula:\[ Z = \frac{X - \mu}{\sigma} \]Substituting \( X = 100 \), \( \mu = 89 \), and \( \sigma = 15 \), we have:\[ Z = \frac{100 - 89}{15} = \frac{11}{15} \approx 0.73 \]

03

Calculate the Probability

Find \( P(Z > 0.73) \) using the standard normal distribution table. Typically, you find \( P(Z < 0.73) \) from the table and subtract it from 1. If \( P(Z < 0.73) \approx 0.7673 \), then\[ P(Z > 0.73) = 1 - 0.7673 = 0.2327 \]Therefore, there is a 23.27% probability that an auction will close above $100.

04

Discuss Bidding Strategy for Part (b)

Setting a maximum bid price too high may result in overpaying, but ensures not missing out due to low bids. Conversely, setting it too low can save money but risks losing the auction. In multiple auctions, higher bid prices increase winning chances across all but may be unnecessary.

05

Determine Percentile for Cutoff in Part (c)

For 10 auctions, a cutoff around the 90th percentile ensures a better chance of winning at least one auction. There is no guarantee of winning as the cutoff only improves odds, not certainty.

06

Find Price for Maximum Bid in Part (d)

Use the 90th percentile of the normal distribution to find the bid price. The Z-score for the 90th percentile is approximately 1.28. Use:\[ X = \mu + Z \times \sigma = 89 + 1.28 \times 15 \approx 108.2 \]So, a bid of around $108 might help win one of the 10 auctions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

A normal distribution, often referred to as a bell curve, is a fundamental concept in statistics. It describes a particular pattern for the distribution of data, where most values tend to cluster around a central mean, with fewer values farther from the mean. In our chemistry textbook auction example, the prices follow a normal distribution with a mean (µ) of \(89 and a standard deviation (σ) of \)15.

To find the probability of a certain event, such as the selling price exceeding $100, a normal distribution allows us to understand how typical or atypical an event is. By converting the price to a standard normal variable, we can compare this to standard normal distribution tables to find the likelihood of different outcomes. This conversion is made using the formula \[ Z = \frac{X - \mu}{\sigma} \]where X is the observed value, `µ` is the mean, and `σ` is the standard deviation.

Bidding Strategy

Understanding how to approach an auction can make a big difference, especially when it comes to setting your maximum bid. A well-thought-out bidding strategy can maximize your chances of winning without overspending.

Consider the following:

• **High Maximum Bid:** Setting your bid too high ensures you won't be outbid, but you may pay much more than the average price, possibly overvaluing the item.
• **Low Maximum Bid:** While this tactic can save money, it risks losing the auction entirely if the item's value exceeds your bid.

When participating in multiple auctions, a higher maximum bid can increase your chances across all auctions, but balance is key to avoid unnecessary expenditure. Tailor your strategy based on how many auctions you're willing to lose and how much money you're willing to spend.

Standard Deviation

Standard deviation is a critical measure when dealing with distributions like the normal distribution. It quantifies the amount of variation or dispersion within a set of data points. In our example of textbook prices, each auction's ending price usually varies around the mean price of $89.

Here, a standard deviation of $15 suggests that most auction prices fall within $15 above or below the mean. The more spread out the data points are, the higher the standard deviation, indicating greater volatility in auction prices.

When determining a winning bid price, taking into account the standard deviation will help you understand the range of likely final prices, allowing for a more informed bidding decision.

Percentiles

Percentiles are useful indicators in statistics to show how a certain value compares to the rest of the data. In the auction scenario, they can help in setting your bid. If you are monitoring 10 auctions, aiming for the 90th percentile bid implies a high chance of winning at least one of the auctions.

A percentile tells you the percentage of data points below a specific value. For example, a 90th percentile bid means that 90% of the auctions close at a price lower than your bid, increasing the probability of a successful bid.

However, it's crucial to remember that while this strategy improves odds, it doesn't guarantee success, especially since auction prices can be unpredictable. Careful calculation and readiness to adjust bids according to auction patterns are essential.