110年 - 110 國立臺灣大學_碩士班招生考試_農業經濟研究所:統計學(E)#102206

(1)One such random sample of 49 bolts produced a mean length of 2.49 inches and a standard deviation of 0.021 inches. Using the 5% significance level, can we conclude that the machine needs to be adjusted? (7%)

(2)Suppose the null hypothesis stated in that example is false and the true mean length of all bolts produced by the machine at the time of the selection of the sample was 2.498 inches. What is the probability of making a Type II error if a=0.05? What is the power of the test? (8%)

(1) Find the p-value for the test of hypothesis with the alternative hypothesis that the current mean monthly salary of all people with a professional degree is greater than $4,961. (8%)

(2) If a=0.01, based on the p-value calculated in part (1), would you reject the null hypothesis? (2%)

(1) Fill the blanks (A)-(L). (6%)

(2) Write the estimated regression equation. (1%)

(3) Explain the meaning of b₂ obtained by estimating the given regression model. (2%)

(4) What are the values of the standard deviation of errors, the coefficient of multiple determination, the adjusted coefficient of multiple determination, SST, SSR, SSE, MSR, and MSE? (3%)

(5) Write the valucs of the standard deviation, the observed value of t for the estimated coefficient b₄.(1%)

(6) What is the predicted value of y for x₁=210, x₂=195, x₃=260, and D₁=1?(1%)

(7) What is the point estimate of the expected (mean) value of y for all elements given that x₁=290, x₂=230, x₃=195,and D₁=0?(1%)

(8) Determine a 99% confidence interval for the coefficient of x₁?(3%)

(9) Using the 2.5% significance level, can you conclude that the coefficient of x₂ is positive? (3%)

(10) Using the 5% significance level, test whether or not the coefficients of all independent variables in the population regression model are equal to zero. (4%)

(1)

(2)

5.(15%) Let X and Y be two independent random variables that follow a Poisson distribution with rate parameter θ and λ, respectively, i.e., X ~ Poisson(θ) and Y ~ Poisson(λ), Show that the distribution of X/(X+Y) is binomial with success probabilityθ/(θ+λ).
Note: the probability density function of a Poisson distribution with rate parameter y is

(1)Find E[Y], Var[Y], and Cov(X, Y). (15%)

(2)Prove that Y/X and Xare independent. (10%)

110年 - 110 國立臺灣大學_碩士班招生考試_農業經濟研究所:統計學(E)#102206

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