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Unformatted text preview: ECO2121: Methods of Economic Statistics Summary from TA8 to TA14 For any random variable X, we are interested in its distribution. If it is parameterized, we thus want to estimate the parameters. To construct C.I.’s or conduct Hypothesis Testing, the key is to get the distribution of the random variables related to the target parameters to be estimated or tested. Note that both C.I.’s and test statistics are random variables. For two random samples of size nX of X ~ N(μX, σX2) and nY of Y ~ N(μY, σY2) independently, see TA8, TA8a, TA13 for details, (a) (b) ~ N(μX, σX2/nX) (no need large n) and (nX – 1)SX2/σX2 = ∑ and SX2 (thus (nX – 1)SX2/σX2) are independent ~ t(nX – 1) ~ χ2(nX – 1) ⁄ (c) Hence, ( − )⁄ The above is similar for Y. (d) ~ F(nX – 1, nY – 1) For “large” sample, first, by CLT, the proportions ̂ ~ N(pX, pXqX/nX) and ̂ ~ N(pY, pYqY/nY); second, t(k) is very “close” to N(0, 1) when the degree of freedom k goes to infinity. Then the C.I.’s for means can be derived accordingly, with different estimators of the corresponding variances,...
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