Question

Problem 4 (Definition of Auction: a public sale in which goods or property are sold to the highest bidder.)
An economist at Vanderbilt University devised a study to compare different types of online auctions. In one experiment he compared a Dutch auction to a first-place sealed bid auction. In the Dutch auction the item for sale starts at a very high price and is lowered gradually until someone finds the price low enough to buy. In the first-price sealed bid auction each bidder submits a single sealed bid before a particular deadline. After the deadline, the person with the highest bid wins. The researcher auctioned off collectible trading cards from the game Magic: The Gathering. He placed pairs of identical cards up for auction; one would go into Dutch auction and the other to the first-price sealed bid auction. He then looked at the difference in the prices he received on the pair. He repeated this for a total of 88 pairs.
[a] Explained why the data should be analyzed using paired samples as opposed to two independent samples.

[b] What makes a pair?

[c] What is the explanatory variable? Is it categorical or quantitative?

[d] What is the response variable? Is it categorical or quantitative?

[e] State the relevant hypotheses in words:
Null hypothesis:
Alternative hypothesis:
[f] Define the parameter of interest and give the symbol that should be assigned to it.

[g] State the relevant hypotheses in symbols (using a parameter):
Null hypothesis:
Alternative hypothesis:

       [h] Assume the p-value is 0.17 (write a conclusion).

Answer 1

• The two cards that are sent are identical, that is, each pair has same characteristics. Sending one card to Dutch auction and the other to first-place sealed bid auction will yeild the same result if the cards are interchanged, or infact, sending a card to Dutch auction once and the same card to first-place sealed bid auction. Hence, due to identical nature of the cards, the data should be analysed as paired samples, and not as independant samples.
  
  
  
• A pair is something that comes in a set of two.The card that is sent to Dutch auction and the one that is send to first-place sealed bid auction together makes one pair. They have similar properties (as they are identical) and hence can be termed as pair.
  
  
  
• The explanatory variable is the characteristic of the card (for example, face value, etc.). This is a categorical variable, as their numerical feature is not studied, but they are used to identify the units. Hence, the explanatory variable is categorical.
  
  
  
• The response variable is the bid amount of the cards. This is a quantitative variable, as as their numerical feature is studied. The quantitative charachteristic is the study variable. That is, using the quantitative nature (bid amount) we set our test statistic. Hence, the response variable is quantitative.
  
  
  
• The Null hypothesis is that, the mean bid amount of the Dutch auction is same as that of a first-place sealed bid auction.
  
  
  The Alternative hypothesis is that, the mean bid amount of the Dutch auction is not equal to that of a first-place sealed bid auction.
  
  
  
  
• We are to test:
  
  where,
  
  
  
  
  
• The p-value is 0.17.
  Since the p-value is greater than 0.05, the Null hypothesis is accepted at 5% level of significance. Hence, it can concluded that the mean of Dutch auction bid is equal to the first-price sealed bid.
  
  
  

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