114年 - 114 國立臺北大學_碩士班一般入學考試試題_統計學系：統計學#147125

1. Let $\{Y1, \ldots, Yn\}$ be a random sample from the normal distribution with mean $\mu$ and known variance $\sigma^2$. The hypotheses are $H0: \mu = 50$ vs $Ha: \mu > 50$ and the rejection region $R = \{\hat{y}: \hat{y} > 50 + a\}$. At 0.05 level of significance, the value of $a$ is (5%) and the power of the test for true $\mu = 60$ is . (5%) (若無法算出一個值，請用累積密度函數等方式表達)

2. The probability density functions for X and (Y given X) are given below.

$$\begin{array}{l} PX(x) = \frac{\lambda^x e^{-\lambda}}{x!} \text{ where } \lambda > 0 \text{ and } x = 1, 2, \ldots \\ P{Y|X}(y|x) = \binom{x}{y} p^y (1-p)^{x-y} \text{ where } 0 < p < 1 \text{ and } y = 1, 2, \ldots \end{array}$$

The probability density function of Y is (5%) and the variance of Y is . (5%)

3. Suppose that each of two investments for a portfolio has a \( 4\% \) chance of a loss of $5 million, a \( 2\% \) chance of a loss of $1 million, and \( 94\% \) chance of a gain of $2 million. Two investments are independent of each other. The 10th percentile for the distribution of this portfolio investment return is __. (5%)

4. Consider a random sample \( \{x_1, \dots, x_n\} \) from the distribution with mean \( \mu \) and known variance \( \sigma^2 \). Define \( \bar{x} = n^{-1} \sum_{i=1}^{i=n} x_i \), \( s_1^2 = (n-2)^{-1} \sum_{i=1}^{i=n} (x_i - \mu)^2 \), and \( s_2^2 = (n-2)^{-1} \sum_{i=1}^{i=n} (x_i - \bar{x})^2 \). If n=60, the distributions of \( (n-2)s_1^2/\sigma^2 \) and \( (n-2)s_2^2/\sigma^2 \) are ________ and ________, respectively. (10%) (Justify your answers)

5. Consider a population of five weights identical in appearance but weighing 1, 3, 5, 7 and 9 ounces. The variance for the population (\( \sigma^2 \)) is ________. (5%) Sampling without replacement from the above population with a sample size of 2 produces ten possible samples. Using the ten sample mean values to calculate the variance of \( \bar{X} \), it is ________. (5%) Why is this value not the same as \( \sigma^2/n \)? ________ (5%)

(1) Conduct a statistical test to determine whether there is a significant difference between the variances in the SAT score of male and female students. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(2) Based on the result of previous question, conduct a statistical test to determine whether the mean SAT score of male students is significantly LOWER than the mean SAT score of female students. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(3) Are the proportions of in-state students the same for male and female students? Report the test statistic and P-value. (5%)

(4) Construct a \( 95\% \) confidence interval for the difference of the proportion of in-state students for male and female students. \( (5\%) \)

(1) Find the within-group variance. (5%)

(2) To test the claim that there is no difference among the means in blood pressure reduction of three groups, what is the test statistics? (5%)

(1) Conduct a test to determine whether there is at least one significant predictor. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(2) Conduct a test to determine whether smoking a significant predictor for the risk of a stroke. Report the test statistic and make a conclusion (reject or not reject the null hypothesis). (5%)

(3) What percentage of the variability in risk of stroke can be explained by the linear effect of the age, blood pressure and he/she is a smoker or not? (5%)

4. Consider an experiment having five possible outcomes whose probabilities are hypothesized to be equal, i.e., 0.2 for all outcomes. This is to be tested by performing 60 independent replications of the experiment. If the resultant number of times that each of the five outcomes occur is 12, 7, 10, 14 and 17, should the hypothesis be rejected? (5%)