100年 - 100 國立交通大學管碩士班考試入學試題_交通運輸研究所_交通所：統計學#124068

1. The probability mass function (pmf) of the binomial distribution is
f(x) =pˣ(1-p)ⁿ⁻ˣ, x = 0,1,2,..., n.
Thus, the moment-generating function, M(t), is _( A )_.

(a) Are X₁ and X₂ dependent or independent? _(B)_.

(b) The covariance of X₁ and X₂ is _(C)_.

3. The distribution of the amount of gravel (in tons) sold by a particular construction supply company in a
given week is a continuous random variable X with probability density function (pdf)

The variance of X, V(X), is _( D )_.

(a) The probability that neither facility is busy more than one-quarter of the time is __(E)__

(b) The marginal pdf of X, which gives the probability distribution of busy time for the drive-up facility
without reference to the walk-up window, is__(F)__.

(a) The joint probability density function (pdf) of (X₁,X₂) is __(G)__.

(b) Let λ₁ = and λ₂ =, so that the expected lifetimes are 1000 hours and 1200 hours,
respectively. The probability that both component lifetimes are at least 1500 hours is __(H)__.

6. Let μ denote the true average tread life of a certain type of tire. Consider testing H₀: μ = 30,000 versus Hₐ: μ ＞ 30,000 based on a sample of size n = 16 from a normal population distribution with standard deviation σ = 1500. A test with significance level α = 0.01 requires Zα = Z0.01 = 2.33. The probability of making a type II error when μ = 31,000 is __(I)__.

(a) The test statistic is__(J)__.

(b) If the results of the experiment yielded y = 1389, and we use a significance level of α = 0.05, is the null hypothesis is accepted or rejected? __(K)__ (Z_α = Z_0.05 = 1.645)

(a) Compute the standard deviations of and . The standard deviation of is (M). The standard deviation of is__(L)__.

(b) Find a 95% confidence interval for the spring constant, . The 95% confidence interval for the spring constant is __(N)__.

(c) Find a 99% confidence interval for the unloaded length of the spring, . The 99% confidence interval for the unloaded length of the spring is __(O)__.

(d) Find a 95% confidence interval of the length of a spring under a load of 1.4lb. The 95% confidence interval of the length of a spring under a load of 1.4lb is__(P)__.

(a) Find the value of k. (5%)

(b) Find the probability that rocks of types 1 and 2 together account for at most 50% of the sample. (5%)

10. Let X₁, X₂, X₃, X₄ be independent random variables that have normal distributions N(μᵢ, σ²),
i = 1, 2, 3, 4. μ_i and σ² are the mean and variance of X_i. In order to test, at the α = 0.05 significance
level, the hypothesis H₀: μ₁ = μ₂ = μ₃ = μ₄ = μ against all alternatives on the basis of a random
sample of size nᵢ = 3 from each of the four distributions. Determine whether we accept or reject H₀ if
the observed values from the four distributions are, respectively, as follows: