109年 - 109 國立臺北大學_碩士班一般入學考試試題_統計學系：統計學#147129

科目：研究所、轉學考(插大)◆統計學 | 年份：109年 | 選擇題數：0 | 申論題數：20

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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 (20)

一、(50%)

1. (10%) Identical twins (同卵雙胞胎) form when a single fertilized egg splits into two genetically identical parts. The identical twins share the same DNA set and are thus always of the same sex. Fraternal (異卵雙胞胎) twins develop when two separate eggs are fertilized and implant in the uterus. The genetic connection is the same as siblings (兄弟姐妹) born at separate times. If 64 percent of all twin pairs are of the same sex, what percentage of twin pairs are identical twins?
2. (10%, 5% each) We are interested in the outcome for rolling dice.

a. Find the sampling distribution when a pair of fair dice is rolled (mean and standard deviation)
b. Determine the variance of the sum obtained when a pair of fair dice is rolled.
3. (10%) The mean of a Poisson random variable X is μ = 4 with standard deviation σ.

Find P(μ - σ < X < μ + σ).
4. (10%, 5% each) If X is a binomial random variable with B(100, 0.1),

a. Find \(\mathrm{P}(12\leq X\leq 14)\)
b. Find the approximate value of \( P(12 \leq X \leq 14) \) using the normal approximation.
5. (10%, 5% each) Let X̄, Ȳ, S²x and S²y be the respective sample means and unbiased estimates of the variances obtained from independent samples of sizes n and m from the normal distributions N(μx, σ²x) and N(μy, σ²y), where μx, μy, σ²x, σ²y are unknown. If the sample sizes n and m are smaller than 30, construct (1 - α)100% confidence interval for μx - μy when

a. two population standard deviations can be assumed equal.
b. two population standard deviations can not be assumed equal.
二、(50%)

1. (15%) Employees of MNM 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.

| Factor A: Program | Factor B: Sex | | Total(y_{i}) |
| --- | --- | --- | --- |
| | Male | Female | |
| Program I | 320 | 380 | 1240 |
| | 240 (560) | 300 (680) | |
| Program II | 160 | 240 | 790 |
| | 180 (340) | 210 (450) | |
| Program III | 240 | 360 | 1270 |
| | 290 (530) | 380 (740) | |
| Total(y_{j}) | 1430 | 1870 | 3300 |

Calculations: ΣΣΣy²φ =969800; Σy²i = 3,774,600; Σy²j = 5,541,800; Σy²k = 1,922,600; SST=62,300



a. State the ANOVA model for the experimental design (i.e. full model of the two-factor analysis of variance).
b. Fit the ANOVA model (i.e. estimate the model parameters).
c. Set up the ANOVA table and test for the significances of factor A, factor B, and interaction of factors A and B at the \(5\%\) level of significance.
2. (10%) A new brand of chocolate bar is being market tested. Four hundred of the new chocolate bars were given to consumers to taste. The consumers were asked whether they liked or disliked the chocolate bar. You are given their responses below.

| Response | Frequency |
| --- | --- |
| Liked | 320 |
| Disliked | 80 |
| | 400 |

a. Construct a \(90\%\) confidence interval for the true proportion of people who liked the chocolate bar.
b. With a 0.90 probability, how large of a sample needs to be taken to provide a margin of error of \(3\%\) or less?
3. (10%) The average starting salary of students who graduated from colleges of Business in 2018 was $38,400. A sample of 100 graduates of 2019 showed an average starting salary of $40,000. Assume the standard deviation of the population is known to be $8000. We want to determine whether or not there has been a significant increase in the starting salaries.

a. State the null and alternative hypotheses. Test the hypothesis at the \(5\%\) level of significance. What do you conclude?
b. Compute the power of the statistical test when \(\mu = 40,000\)
4. (15%) Given below are five observations collected in a regression study on two variables, x (independent variable) and y (dependent variable).

| x | y |
| --- | --- |
| 2 | 4 |
| 3 | 4 |
| 4 | 3 |
| 5 | 2 |
| 6 | 1 |

$$\bar{x} = 4, \bar{y} = 2.8, S{xx} = 10, S{yy} = 6.8, S_{xy} = -8$$

a. Develop the least squares estimated regression equation.
b. At \(95\%\) confidence, perform a \(t\) test and determine whether or not the slope is significantly different from zero.
c. Perform an \(F\) test to determine whether or not the model is significant. Let \(\alpha = 0.05\).
d. Compute the coefficient of determination. Explain it.
e. Construct a \(95\%\) confidence interval and prediction interval of \(y\) with \(x = 5\)