Question

2. Significance tests

Buying an item sight unseen on the Internet requires a significant amount of trust in the seller. Consider this hypothesis: Potential buyers tend to scrutinize the offers posted by sellers with poor reputations more than they do the offers posted by sellers with neutral or good reputations. As a result, if buyers notice a surcharge (such as a shipping fee) levied by a seller with a poor reputation, they reduce the (presurcharge) price they are willing to pay for the item. On the other hand, a surcharge does not affect buyers’ (presurcharge) willingness to pay for an item offered by a seller with a neutral or a good reputation.

Amar Cheema tested this hypothesis, which was described in a June 2008 paper entitled “Surcharges and Seller Reputation” and published in the Journal of Consumer Research. Cheema collected data on 271 completed eBay auctions for three DVD trilogies: The Godfather, The Lord of the Rings, and Star Wars. For each auction, Cheema recorded the winning bid, the surcharge, and the seller’s eBay feedback score. Then he partitioned the 271 auctions into three almost equal-sized samples based on the seller’s feedback score.

The following is a simple linear regression model estimated for each group:

yy	=  =	β0+β1x + ε,β0+β1x + ε,
where ywhere y	=  =	winning bid (in dollars), andwinning bid (in dollars), and
xx	=  =	shipping cost (in dollars).shipping cost (in dollars).

The following equation lists the estimation results obtained for the sample of 90 high-reputation sellers:

The estimated regression equation:The estimated regression equation:		ŷ=29.95−0.35xŷ=29.95−0.35x
SSR:SSR:		5050
SSE:SSE:		5,5005,500

(Note: These results do not exactly duplicate Cheema’s results but are representative of the Cheema study.)

The mean square due to error (MSE) s² is an unbiased estimator of σ², the variance of the error variable ε in the regression model. In this regression analysis, the MSE equals , and the standard error of estimate equals .

Hint: For the next question, ∑(xi - x̄)2xi - x̄2= (n – 1)(sample variance of shipping cost). The sample variance of shipping costs for the auctions in the sample is 4.54.

A different sample of eBay auctions cannot be expected to provide the same value of b₁ as the current sample. So b₁ is a random variable. Its sampling distribution has an estimated standard deviation of .

Use the Distributions tool to help you answer the questions that follow.

t Distribution

Degrees of Freedom = 80

-3-2-10123x

The regression model specifies that winning bid and shipping cost are linearly related. Conduct a t test (at the significance level α = 0.05) for a significant linear relationship between shipping cost and winning bid when the seller has a high reputation.

The value of the test statistic is , and you conclude that there is a significant linear relationship between shipping cost and winning bid. The result of the significance test consistent with the notion that buyers don’t pay attention to a surcharge when it is levied by a high-reputation seller.

The 95% confidence interval estimate of β₁ is to .

Since 0 is   in the confidence interval, you conclude that there is a significant linear relationship between shipping cost and winning bid. This result consistent with the notion that buyers don’t pay attention to a surcharge when it is levied by a high-reputation seller.

Conduct an F test (at a significance level α = 0.05) of overall significance for the regression. (With 1 degree of freedom in the numerator and n – 2 = 90 – 2 = 88 degrees of freedom in the denominator, F = 3.949 provides an area of 0.05 in the upper tail.)

The test statistic is , and you conclude that there is a significant linear relationship between shipping cost and winning bid.

Answer 1

In this regression analysis, the MSE = SSE / (n-2) = 5500/(90-2) = 62.5

The standard error of estimate = = 7.905694 .

SSxx = (n-1) = (90 - 1) * 4.54 = 404.06

A different sample of eBay auctions cannot be expected to provide the same value of b₁ as the current sample. So b₁ is a random variable. Its sampling distribution has an estimated standard deviation of  

standard error of estimate / = 7.905694 / =  0.3932938

Critical value of t at df = 90-2 = 88 and 0.05 significance level is 1.99. The slope coefficient is significant if t statistic is less than -1.99 or greater than 1.99.

The value of the test statistic is = Slope / standard deviation =  -0.35 / 0.3932938 = -0.88992

We conclude that there is no significant linear relationship between shipping cost and winning bid.

The 95% confidence interval estimate of β₁ is

-0.35 - 1.99 * 0.3932938 to  -0.35 + 1.99 * 0.3932938

-1.132655 to 0.4326547

Since 0 is in the confidence interval, you conclude that there is no significant linear relationship between shipping cost and winning bid.

MSR = SSR / (k - 1) = 50 / (2 -1) = 50

The test statistic is F = MSR / MSE = 50 / 62.5 = 0.8 ,

Since F statistic is less than the critical value of 3.949, we conclude that there is no significant linear relationship between shipping cost and winning bid.