Albury and Wodonga are two adjacent towns on the border between New South Wales and Victoria. Albury is in NSW and Wodonga is in VIC. We have the selling price and the location of the last 32 properties sold in Albury-Wodonga. We have generated the binary variable $W$, which is equal to 1 if the property is in Wodonga, and is equal to 0 otherwise. It is believed that people prefer to locate in NSW than in VIC, and as a result the mean of house prices in Albury is higher than the mean of house prices in Wodonga. The null and the alternative that would allow us to test this belief in the model: $$ price_i = \beta _0 + \beta _1 W_i + u_i $$ are:

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BlackTom AI · Accepted Answer

First, restating the setup helps orient the reasoning: we have a simple linear model price_i = beta0 + beta1 W_i + u_i, where W_i = 1 if the property is in Wodonga (VIC) and W_i = 0 if it is in Albury (NSW). The substantive claim is that people prefer NSW, so the average price in NSW (W_i = 0, i.e., Albury) is higher than in VIC (W_i = 1, i.e., Wodonga).

Option under consideration: b. H0: beta1 = 0 against H1: beta1 < 0.

Reasoning about what beta1 represents: in this model, beta1 is the difference in mean price between properties in Wodonga and properties in Albury. Specifically, E[price | W=1] = beta0 + beta1 and E[price | W=0] = beta0. Therefore, beta1 = E[price | W=1] − E[price | W=0]. If prices in Albury (W=0) are higher than in Wodonga (W=1), then E[price | W=1] would be smaller, making beta1 negative.

Now evaluate the null and alternative: H0: beta1 = 0 posits no difference in mean prices between the two towns. H1: beta1 < 0 posits that the mean price in Wodonga (VIC) is lower than in Albury (NSW), i.e., the difference E[price | W=1] − E[price | W=0] is negative, which matches the belief that NSW locations yield higher prices.

Why this choice aligns with the stated belief: since the hypothesis specifies a one-sided test (beta1 < 0), it directly tests whether the Wodonga mean is strictly smaller than the Albury mean, consistent with the claim that people prefer NSW and thus that NSW prices are higher.

Why an alternative (two-sided) test would be inappropriate here: a two-sided test H0: beta1 = 0 vs H1: beta1 ≠ 0 would test for any difference without direction, which is unnecessarily broad given the belief specifying a direction (NSW higher, VIC lower). If the data happened to show VIC prices higher, a two-sided test would still reject H0, but that would be contrary to the stated belief; thus the one-sided formulation is more appropriate.

In summary, the presented option captures exactly the directional hypothesis implied by the belief about price differences between Albury and Wodonga, by testing whether the Wodonga coefficient beta1 is less than zero rather than equal to zero or greater than zero.

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