91Ó°ÊÓ

The textbook you need to buy for your chemistry class is expensive at the college bookstore, so you consider buying it on Ebay instead. A look at past auctions suggest that the prices of that chemistry textbook have 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

We're dealing with a normal distribution problem where auction prices for a textbook are normally distributed with a mean of $89 and a standard deviation of $15. We need to find the probability of prices exceeding $100.

02

Standardizing the Value

To find the probability of an auction closing at more than \(100, we convert \)100 to a z-score using the formula: \( z = \frac{X - \mu}{\sigma} \), where \( X = 100 \), \( \mu = 89 \), and \( \sigma = 15 \).

03

Calculate the Z-Score

Substitute the values into the formula: \( z = \frac{100 - 89}{15} = \frac{11}{15} \approx 0.73 \).

04

Probability Calculation

Using a standard normal distribution table or calculator, find the probability of \( z > 0.73 \). This corresponds to \( 1 - P(Z < 0.73) \approx 0.233 \). So, the probability that an auction closes above $100 is approximately 23.3%.

05

Analyze Bid Price Strategies

Setting a bid too high can result in overpaying; setting it too low might mean missing out on winning any auctions. With multiple auctions, a higher maximum bid increases the probability of winning but at a riskier price.

06

Determine Percentile for Winning

For 10 auctions, you might consider using around the 90th percentile as your maximum bid to be 'somewhat sure' to win one, but there is no guarantee of winning due to variability in auction results. The 90th percentile can be found using the formula: \( P(Z < x) = 0.9 \).

07

Calculate Desired Maximum Bid Price

If \( Z \approx 1.28 \) for the 90th percentile, use \( X = \mu + Z\sigma = 89 + 1.28 \times 15 \approx 108.2 \). Your bid price might be around $108.2 to increase the likelihood of winning.

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Key Concepts

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

Z-score

The concept of a Z-score is a fundamental topic in statistics, particularly in the context of normal distribution. A Z-score essentially tells us how many standard deviations an element is from the mean. For our problem, we're trying to find out the probability that an auction exceeds \(100.
This involves standardizing our auction price using the Z-score formula:

• \( z = \frac{X - \mu}{\sigma} \)

where:

• \( X \) is the value of interest (in this case, \)100),
• \( \mu \) is the mean (\(89),
• \( \sigma \) is the standard deviation (\)15).

Substituting these into the formula gives us a Z-score of approximately 0.73.
This number provides a way for us to move to the next step: determining how likely it is for this event to occur by looking it up in a standard normal distribution table.
The Z-score helps us compare our situation to a standard normal distribution and thus enables us to calculate probabilities.

Probability Calculation

After determining the Z-score, we move to probability calculation. This step involves finding out the likelihood of the auction price exceeding \(100.
Using Z-score tables or statistical software, we find the probability for our calculated Z-score of 0.73. Essentially, we are looking for:

• \( P(Z > 0.73) \)

This is equivalent to:

• \( 1 - P(Z < 0.73) \approx 0.233 \)

meaning there is about a 23.3% chance that the auction price will exceed \)100.
This probability tells us how realistic it is to expect prices above a certain threshold and plays a crucial role in decision-making strategies, as it informs the buyer about the chances of encountering higher-priced auctions.

Auction Strategies

In an auction scenario like the one described, understanding the strategy behind your bids is key. One common strategy is manipulating your bid to increase the chances of winning without overpaying.
When setting a bid price, several factors come into play:

• A very high bid might ensure a win, but you could overpay significantly, especially if other bids are lower. This is not ideal for single auctions.
• A very low bid could result in losing the auction, which isn't beneficial if you need the item.
• If participating in multiple auctions, setting a higher bid may enhance your chances of success across the board but comes with the risk of higher final costs.

For someone watching several auctions, calculating the 90th percentile could be advisable to be 'somewhat sure' of winning one.
This requires understanding auction variability and deciding on a bid price that sits comfortably within your financial limits.
For instance, a 90th percentile with a bid at about $108.2 provides a balance, offering a reasonable chance of winning while controlling potential overbidding.