97年 - 97 國立交通大學管碩士班考試入學試題_運輸科技與管理學系：統計學#124784

(a) X and Y are independent chi-square random variables with 2 and 5 degrees of freedom, respectively. What kind of distribution the random variable X+Y will be? (6%)

(b) What is the meaning of adjusted R² for the regression analysis?(6%)

(c) Let the variables X₁, X₂, and X₃ have the variances σ₁² = 1.0, σ₂² = 3.0, σ₃² = 5.0. Let these variables be independent and Y = 13 - 2X₁ + 3X₂ - 10X₃. Find the variance of Y.(6%)

(d) Let Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ be a multiple regression model with X₃ =.
What would be happened when we try to estimate the parameters of this model? and explain
why? (6%)

(e) What assumptions are usually made about the error terms of linear regression models? (6%)

2. (10%) The annual rainfall (in inches) in a certain region is normally distributed with μ = 30, σ = 3. What is the probability that in 2 of the next 4 years the rainfall will exceed 35 inches? Assume that the rainfalls in different years are independent.

3. (10%) If the density function of X equals

Please find the value of c. What is probability for X>2 (i.e. P(X>2))?

(a) What is the distribution of ? (5%)

(b) If= 4, give an interval that, with 90 percent confidence, will contain the value of
Xₙ₊₁. (5%)

5. (10%) A simple random sample of 500 students from a university was selected to estimate the mean living expense during last academic year. Another simple random sample of 550 students was selected, independently of the first, to estimate the mean living expense during this academic year. The sample results (expressed in thousand dollars) were as follows:

Construct a 95% confidence interval for μ₂ - μ₁. Can one conclude that the mean living expense  per student has changed between the two years at α = 0.05?

(a) Estimate the components of variance, within and between groups, and the total mean. (5%)

(b) Compute the analysis of variance table, including the expected mean squares. (5%)

(c) Is there a significant between worker variation at α = 0.05. (5%)

7. (15%) In studying the accidents occurred on an expressway, a consultant recorded the number of accidents occurred each day in a random sample of 400 days. The sample results follow:
The consultant wishes to determine whether the probability distribution of the number of accidents occurred in one day is Poisson. [i.e., P(x) = ]. Use the chi-square procedure to conduct the test, controlling the α risk at 0.10. Pool classes as required. State the alternatives,the decision rule, the value of the test statistic, and the conclusion.
[i.e., Exponential distribution: f(x) = λexp(-x)； F(x) = P(X ≤ x) = 1-exp(-λx) ]