MA4114 HW9 (1) A company held a raffle during an event

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MA4114 HW9 (1) A company held a raffle during an event. The designer of that raffle claimed that 1% of ticket holders would be awarded the grand prize, 5% would be awarded second prize, 10% dould be awarded the third prize, and the rest would get no prize. Of the 2000 people who got a raffle ticket, 30 won the grand prize, 120 won the second prize, 210 won the third prize, and the rest got nothing. Use a chi-square goodness-of-fit test to test if the model the designer described is a good fit. Use the significance level 0.05. (2) The following data shows a sample of students who applied to one of the four doctoral programs at a university. Acceptance Program Not Accepted Accepted Total Education 13 26 39 English 37 72 109 Pharmacy 78 157 235 Psychology 15 21 36 Total 143 276 419 (a) Test the hypothesis that the acceptance rate for doctoral programs is independent of the programs they apply to. Use α = 0.1. (b) Create a 95% confidence interval for the overall acceptance rate for doctoral programs at the university. (3) A study was done to study the effect of ambient temperature x on the electric power consumed by a chemical plant y. Other factors were held constant, and the data were collected from an experimental pilot plant. x (? F) y (BTU) 27 250 45 285 72 320 58 295 31 265 60 298 34 267 74 321 (a) Estimate the slope (β) and intercept (α) of the least squares regression model. (b) Predict the power consumption for an ambient temperature of 65 degreess Fahrenheit. (c) Evaluate s2 = SSE . n−2 1 2 (d) Test the hypothesis H0 : β = 0 vs. H1 : β 6= 0 at α = 0.05 level of significance and interpret the resulting decision. (e) Find the coefficient of determination and interpret the meaning. (f) Construct a 95% confidence interval for β. (g) Construct a 95% confidence interval for the mean power consumption when x = 65? F . (h) Construct a 95% prediction interval for a single predicted value of power consumption when x = 65? F . (4) Given the two random variables X and Y that have the joint density f (x, y) = x · e−x(1+y) for x > 0 and y > 0 find the regression equation of X on Y . Sketch the regression curve. (5) Given the joint density f (x, y) = 6x for 0 < x < y < 1 find µY |x and µX|y .