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科目：研究所、轉學考(插大)◆統計學 | 年份：115年 | 選擇題數：0 | 申論題數：19

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所屬科目：研究所、轉學考(插大)◆統計學

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申論題 (19)

1. (10%) Two cards are randomly chosen without replacement from an ordinary deck of 52 cards. Let B be the event that both cards are aces, let A_s be the event that the ace of spades is chosen, and let A be the event that at least one ace is chosen. Find

(a) \((5\%)\) P(B|A_s)
(b) \((5\%)\) P(B|A)
2. (15%) Consider the experiment of tossing a fair coin three times. Let X denote the number of heads on the first two tosses and Y denote the number of heads on all three tosses.

(a) \((6\%)\) Find the joint probability mass function (pmf) of \((X,Y)\).
(b) \((4\%)\) Find the marginal probability mass function (pmf) and the cumulative distribution function (cdf) of X.
(c) \((5\%)\) Compute the correlation coefficient of X and Y.
3. (15%) Let X is an exponential random variable with rate λ. Let a denote a constant, a > 0, and Y = ae^X

(a) \((5\%)\) Find the cumulative density function (cdf) of Y.
(b) \((2\%)\) Find the probability density function (pdf) of Y.
(c) \((8\%)\) Find E(Y) and Var(Y)
4. (10%) China Airline quote a flight time of 2 hours, 5 minutes for its flights from Taipei to Shanghai. Suppose we believe that actual flight times are uniformly distributed between 2 hours and 2 hours, 20 minutes.

(a) \((5\%)\) What is the probability will be more than 10 minutes late?
(b) \((5\%)\) What are the mean and variance of the flight times?
5. (10%) The following results come from two independent random samples taken of two different normal populations. Assume that two variances are unknown, but they are the same.



(a) \((5\%)\) What is the point estimate of the difference between the two population means?
(b) \((5\%)\) Find a \(95\%\) confidence interval for the different between the two population means.
6. (10%) The following table contains observed frequencies of a sample of 500. Test for independence of the row and column variables using α=0.01 level of significance.
7. (10%) In an experiment to investigate the performance of four different brands of tires for 125-cc motorcycles, five tires of each brand were tested and number of miles until failure was observed. A partially completed ANOVA table is given:

| Source of variation | Sum of Squares | Degrees of Freedoms | Mean Square | F-value |
| --- | --- | --- | --- | --- |
| Treatments | (?) | (?) | (?) | (?) |
| Error | 235419 | (?) | (?) | |
| Total | 310501 | (?) | | |
The following table is the cumulative probabilities for the standard distribution



Notes: The attached probability value is the probability in the upper tails of Chi_square ($\chi^2$), or t_distribution or F distribution.

$$t_{0.025}(23)=2.069 \quad t_{0.05}(23)=1.714 \quad t_{0.025}(24)=2.064 \quad t_{0.05}(24)=1.711$$

$$\chi^2_{0.05}(3) = 7.815 \quad \chi^2_{0.01}(3) = 11.345 \quad \chi^2_{0.05}(4) = 9.488 \quad \chi^2_{0.01}(4) = 13.277 \quad \chi^2_{0.025}(4)=11.143$$

$$F_{0.05}(16,3) = 8.69 \quad F_{0.05}(3,16) = 3.24 \quad F_{0.025}(3,16) = 4.08$$

$$F_{0.05}(1,3) = 10.13 \quad F_{0.05}(3,1) = 251.71 \quad F_{0.025}(3,1) = 864.15$$



(a) \((7\%)\) Complete the ANOVA table.
(b) \((3\%)\) At the \(\alpha = 0.05\) level of significance, test whether the average number of miles until failure for the four different brands of tires are equal.
8. (20%) Given are following observations collected in a regression study on two variables. Consider the linear regression model
$$Y = \beta_0 + \beta_1 X + \varepsilon$$

| $$x_t$$ | 2 | 3 | 4 | 5 | 6 |
| --- | --- | --- | --- | --- | --- |
| $$y_t$$ | 3 | 7 | 5 | 11 | 14 |

(a) \((5\%)\) Find the estimated regression equation for these data.
(b) \((5\%)\) Use the estimated regression equation to predict the value of \(y\) when \(x = 7\).
(c) \((5\%)\) Find the coefficient of determination \(R^2\)
(d) (5%) Use F test to test the hypotheses \( H_0 \): \( \beta_1 = 0 \) vs. \( H_a \): \( \beta_1 \neq 0 \) at 0.05 level of significance.