Question

Assume that you are hired by the Governor’s Office to study whether a tax on liquor has affected alcohol consumption in the state. You are able to obtain, for a sample of fifty individuals selected at random, the difference in liquor consumption (in ounces) for the year before and after the tax. For person i who is sampled randomly from the population, Yi denotes the change in liquor consumption. Treat these as a random sample from a Normal(µ, σ2 ) distribution.

(A) The Governor believes there was no change in alcohol consumption. How do you test his claim? State the null and alternative hypotheses.

(B) What if the Governor thought there was actually a drop in liquor consumption as a result of the tax? How would you setup the test in this case? State the null and alternative hypotheses. What economic arguments would you use to justify your choice of hypothesis test? 4

(C) Now assume that for your sample of size n=50, you have obtained a mean ?̅ = −32.8 and a sample standard deviation s = 100. Formally conduct the hypothesis test from part (B) using these values and a 5% significance level. Does your conclusion change at a 1% significance level?

(D) Construct a 95% confidence interval for the mean change in liquor consumption based on your hypothesis from point (A).

Answer 1

(A)

The Governor thought there was no change in alcohol consumption. We assume the null hypothesis to be that there was actually no increase in liquor consumption as a result of the tax and if we are able to reject the null hypothesis , we can reject the Governor's claim.

Null Hypothesis H0: Mean difference in liquor consumption (in ounces) for the year after and before the tax is zero.

Alternative Hypothesis Ha: Mean difference in liquor consumption (in ounces) for the year after and before the tax is not equal to zero.

(B)
The Governor thought there was actually a drop in liquor consumption as a result of the tax. We assume the null hypothesis to be that there was actually an increase in liquor consumption as a result of the tax and if we are able to reject the null hypothesis , we can warrant the Governor's claim.

Null Hypothesis H0: Mean difference in liquor consumption (in ounces) for the year after and before the tax is greater than or equal to zero.

Alternative Hypothesis Ha: Mean difference in liquor consumption (in ounces) for the year after and before the tax is less than zero.

(C)

Standard error of mean = s / = 100 / = 14.14214

Test statistic, t = (-32.8 - 0) / 14.14214 = -2.32

Degree of freedom = n-1 = 50-1 = 49

P-value = P(t < -2.32) = 0.0123

Since, p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is strong evidence that difference in liquor consumption (in ounces) for the year after and before the tax is less than zero at 5% significance level.

Since, p-value is greater than 0.01 significance level, we fail to reject null hypothesis H0 and conclude that there is no strong evidence that difference in liquor consumption (in ounces) for the year after and before the tax is less than zero at 1% significance level.

(D)

Z value for 95% confidence interval is 1.96

95% confidence interval for the mean change in liquor consumption based on your hypothesis is,

(-32.8 - 1.96 * 14.14214,  -32.8 + 1.96 * 14.14214)

(-60.52 , -5.08)