108年 - [無官方正解]108 國立臺北大學_碩士班一般入學考試試題_國際企業研究所：統計學#147892

科目：研究所、轉學考(插大)◆統計學 | 年份：108年 | 選擇題數：10 | 申論題數：20

試卷資訊

所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (10)

1. In order not to violate the requirements necessary to use the chi-square distribution, each expected frequency in a goodness of fit test must be

(A) at least 5
(B) at least 10
(C) no more than 5
(D) less than 2
2. The sampling distribution for a goodness of fit test is the

(A) Poisson distribution
(B) t distribution
(C) normal distribution
(D) chi-square distribution
3. An ANOVA procedure is applied to data obtained from 6 independent samples where each sample contains 20 observations. The degrees of freedom for the critical value of F are

(A) 6 numerator and 20 denominator degrees of freedom
(B) 5 numerator and 20 denominator degrees of freedom
(C) 5 numerator and 114 denominator degrees of freedom
(D) 6 numerator and 20 denominator degrees of freedom
4. In an analysis of variance problem involving 3 treatments and 10 observations per treatment, SSE = 399.6. The MSE for this situation is

(A) 133.2
(B) 13.32
(C) 14.8
(D) 30.0
5. In regression analysis, the unbiased estimate of the variance is

(A) coefficient of correlation
(B) coefficient of determination
(C) mean square error
(D) slope of the regression equation
6. In simple linear regression analysis, which of the following is not true?

(A) The \(F\) test and the \(t\) test yield the same conclusion.
(B) The \(F\) test and the \(t\) test may or may not yield the same conclusion.
(C) The relationship between X and Y is represented by means of a straight line.
(D) The value of \( F = t^2 \).
7. In a regression and correlation analysis if r² = 1, then

(A) SSE = SST
(B) SSE = 1
(C) SSR = SSE
(D) SSR = SST
8. In order to determine whether or not the means of two populations are equal,

(A) a t test must be performed.
(B) an analysis of variance must be performed.
(C) either a t test or an analysis of variance can be performed.
(D) a chi-square test can be performed.
9. In a regression model involving more than one independent variable, which of the following tests must be used in order to determine if the relationship between the dependent variable and the set of independent variables is significant?

(A) t test
(B) F test
(C) Either a t test or a chi-square test can be used.
(D) chi-square test
10. In regression analysis, which of the following is not a required assumption about the error term?

(A) The expected value of the error term is one.
(B) The variance of the error term is the same for all values of X.
(C) The values of the error term are independent.
(D) The error term is normally distributed.

申論題 (20)

1.1 (10%) Let the random variable X have the pmf $$f(x) = \frac{(|x| + 1)^2}{9}$$, $$x = -1, 0, 1$$

(a) \((5\%)\) Find \(E(X)\).
(b) \((5\%)\) Find \(E\left(3X^{2} - 2X + 4\right)\).
1.2 (10%) It is claimed that 15% of the ducks in a particular region have patent schistosome infection. Suppose that seven ducks are selected at random. Let X equal the number of ducks that are infected.

(a) \((5\%)\) Find \(\mathrm{P}(\mathrm{X} = 1)\)
(b) \((5\%)\) Find \(\mathrm{P}(\mathrm{X}\geq 2)\)
1.3 (10%) If X is normally distributed with a mean of 77 and a variance of 25. Let $$\overline{X}$$ be the mean of a random sample of size 16 from this distribution.

(a) \((5\%)\) Find \(P(77 < X < 79.5)\).
(b) \((5\%)\) Find \(P\left(77 < \overline{X} < 79.5\right)\).
1.4 (10%) Suppose the population is normally distributed with unknown standard deviation $$\sigma$$. Consider the following random sample of 4 observations: 25, 47, 32, 56.

(a) \((5\%)\) Find a point estimate of \(\sigma\)
(b) \((5\%)\) Find a \(95\%\) confidence interval for \(\mu\).
1.5 (10%) Let X and Y denote the tarsus lengths of male and female grackles, and $$\mu_X$$ and $$\mu_Y$$ are the male and female mean tarsus lengths, respectively. Two samples of size 25 are taken from each populations assumed to be normally with equal variances. The male sample has a mean of 35.5 and standard deviation 3.0 while the female sample has a mean of 33.0 and standard deviation 4.0. We want to test the hypothesis that the mean tarsus lengths are equal against the hypothesis that male has longer tarsus lengths.

(a) \((2\%)\) Write the null and alternative hypotheses with symbols.
(b) \((6\%)\) Compute the test statistics.
(c) \((2\%)\) Use 0.05 as significance level and draw a conclusion.
2.2 (10%) Among 100 managers with degrees in business administration, the following data have been accumulated as to their fields of concentration.

| Major | Position | | TOTAL |
| --- | --- | --- | --- |
| | Top Management | Middle Management | |
| Management | 28 | 22 | 50 |
| Marketing | 12 | 8 | 20 |
| Accounting | 10 | 20 | 30 |
| TOTAL | 50 | 50 | 100 |

To test if the position in management is independent of field (major) of concentration,

(a) (5%) State the null and alternative hypotheses.
(b) (5%) At 95% confidence using the p-value approach, test the hypotheses. What is your conclusion? (For the test, give the null and alternative hypotheses, test statistic, and conclusion.)
2.3 (10%)Employees of AAA Corporation are about to undergo a retraining program. Management is trying to determine which of three programs is the best. They believe that the effectiveness of the programs may be influenced by sex. A factorial experiment was designed. You are given the following information of score.

Factor B (Sex)

| Factor A (Program) | 1 | 2 |
| --- | --- | --- |
| A | 26 19 20 21 | 24 21 20 23 |
| B | 30 24 25 29 | 33 27 31 29 |
| C | 26 22 27 17 | 31 23 24 26 |

Given SSA=184.33, SSB=28.17, SSAB=8.33, SST=405.83

(a) (5%) Set up the ANOVA table.
(b) (5%) What advice would you give AAA Corporation? Use
a .05 level of significance. (Hint: Test for any significant differences due to programs, sex, and interaction. For each test, give the null and alternative hypotheses, test statistics, and conclusion.)

2.4 (10%) Multiple regression analysis was used to study how an individual's income (Y in thousands of dollars) is influenced by age (X₁ in years), level of education (X₂ ranging from 1 to 5), and the person's gender (X₃ where 0=female and 1=male). The following is a partial result of computer output that was used on a sample of 20 individuals.

| ANOVA | | | | |
| --- | --- | --- | --- | --- |
| | df | SS | MS | F |
| Regression | | 84 | | |
| Residual | | 112 | | |
| Predictor | Coefficients | Standard Error of Coefficient | | |
| Constant | 21.00 | 0.70 | | |
| X₁ | 0.62 | 0.10 | | |
| X₂ | 0.92 | 0.19 | | |
| X₃ | -0.51 | 0.92 | | |

(a) \((2\%)\) Present the estimated regression equation and compute the coefficient of determination. Explain it.
(b) (2%) Use the \(t\) test to determine the significance of each independent variable. Let \(\alpha = 0.05\). (For each test, give the null and alternative hypotheses, test statistic, and conclusion.)
(c) \((2\%)\) Use the \(F\) test to determine the whether or not the regression model is significant. Let \(\alpha = 0.05\). (For the test, give the null and alternative hypotheses, test statistic, and conclusion.)
(d) (2%) Does the estimated regression equation provide a good fit for the observed data? Explain it based upon the results from part (a), (b), and (c).
(e) (2%) Suppose a new person with \( X_{1} = 40 \), \( X_{2} = 4 \), \( X_{3} = 1 \). Use the estimated regression equation from part (a) to estimate the new person's income.