110年 - 110 國立臺灣科技大學_碩士班招生試題_工業工程系：統計學#111373

(a) If 5 passengeru are selected at random from country C pasengers, ind the probability that there is exactly 1 pussenger carrying COVID-19 virws. (5 points)

(b) If a pasienger is selected at randoun from the fight, find the probability that this pasenger is carrying COVID-19 virus.(5 points)

(c) If 5 passengerw are selected a at random from the fight, find the probability that there is exactly 1 pasenger carrying COVID-19 virus. (5 points)

(d) Suppoee that a passenger is selected at random from the fight and is found to carry COVID-19 virw. Find the probability that the passenger is from country C. (5 points)

(a) Find the joint probability distribution of X and Y. (5 points)

(b) Find the probability P{Y = 1}. (5 points)

(c) Find the conditional distribution of X, given that Y -1. (5 points)

(d) Find the probability P{X +Y≥2}. (5 points)

3. (10 points) If the joint density function of two random variables X and Y is given by f(x，y) = 2 for 0 ＜x≤y ＜ 1. What is the corre. elation coefielent of X and Y?

(a) Pleme test whether the variances of the time to failure of the parts produced by different suppliers are equal, (Please clearly state the null, the alternative hypothesis, and the test statistic, and use a= 0.10) (5 pointa)

(b) The compuny has decided that if the m longer than it is for the current supplier, it will switch suppliers. Should the company switch mean time to failure for the new su uppliers is signifcantly suppliers? (Please clearly state the null, the alternat use a= 0.05) (10 points) ative bypothesis, and the test statistic, and

(a) Please construct the ANOVA table. (10 points)

(b) Please state the null and alternative bypothesis, perform the test, and state your conclusion using 0.05 level of significance. (5 pointe)

(a) Plot the least squares S prediction equation associated with two different neighborboods, and discuss your findings. (5 points)

(b) Test the overal utility of the model (Please clearly state the null, the alernative hypothesis, and the test statistic, and use a = 0.05), (5 points)

7. (10 points) Let X₁,X₂,..., X_n denote a a random sample from a distribution that is N(θ, 1), where the mean θ is unknown wn. Please show that there is no uniformly most powerful (UMP) test of the simple hypothesis H₀: θ= θ₀, where lo be a specified value of θ, against the alternative hypothesis

H1:θ≠θ₀.

110年 - 110 國立臺灣科技大學_碩士班招生試題_工業工程系：統計學#111373

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