# Problem Set 6 Statistics Worksheet

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Problem Set 6 Statistics Timothy Roeper Due March 28, 2021 1. You are hired by the governor of New York to study whether a tax on liquor has decreased average liquor consumption in your state. You are able to obtain, for a sample of individuals selected at random, the difference in liquor consumption (in ounces) for the years before and after the tax. For person i who is sampled randomly from the population, Yi denotes the change in liquor consumption. Suppose we know that the variance of the decrease in liquor consumption is 400 (σ 2 = 400). We do not know the mean µ of the decrease in liquor consumption and we estimate it using a sample of 400 people. We estimate µ using Ȳ which is based on the sample of 400 people: P400 Yi Y1 + Y2 + Y3 + . . . Y400 Ȳ = = i=1 400 400 (a) Suppose the liquor tax actually had no impact on consumption (µ = 0), what is the probability of finding a Ȳ of 1.5 ounces or more in your sample (Find P (Ȳ ≤ −1.5|µ = 0)). Answer We know that Ȳ will be approximately normally distributed (Central Limit Theorem). We therefore can derive its probability distribution by finding its expected value and variance. If we assume that the liquor tax had no impact, that means µ = 0, so E[Ȳ ] = µ = 0 Already stated in the question is that the variance of a randomly drawn person, σY2i , is 400. Therefore, by the law of large numbers, the variance of Ȳ will be the variance of the population divided by the sample size: σȲ2 = σY2i =1 400 And ...
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