# 統計學的問題 關於母體樣本

1. Suppose that x1 and x2 are random sample of observations from a population with mean u(母體平均數) and variance s平方. Consider the following three point estimators,X,Y,Z,of u(母體平均數):

X=1/2 x1 + 1/2 x2 (大寫X 跟小寫x)

Y=1/4 x1 + 3/4 x2(大寫X 跟小寫x)

Z=1/3 x1 + 2/3 x2(大寫X 跟小寫x)

a. Show that all three estimators are unbiased.

b. Which of the estimators is the most efficient?

c. Find the relative efficiency of X with respect to each of the other two estimators.

2. There is concern about the speed of automobiles traveling over a particular of highway. For a random sample 28 automobiles,radar indicated the following speeds,in miles per hour:

59 63 68 57 56 71 59

69 53 58 60 66 51 59

54 64 58 57 66 61 65

70 63 65 57 56 61 59

a. Check for evidence of nonnormality.

b.Find a point estimate of the population mean that is unbiased and efficient.

c.Use an unbiased estmation procedure to find a point estimate of the variance of the sample mean.

3.A random sample of 10 economist produced the following forecasts for percentage growth in real domestic product in the next year:

2.2 2.8 3.0 2.5 2.4 2.6 2.5 2.4 2.7 2.6

Use unbiased estimation procedures to find estimate for the following:

a.The population mean

b.The population variance

c.The variance of the sample mean

d. The population proportion of economists predicting growth of at least 2.5% in real gross domestic product

4.Twenty people in one large metropo;itan area were asked to record the time (in minutes) that it takes them to drive to work. These were as follow :

30 42 35 40 45 22 32 15 41 45

28 32 45 27 47 50 30 25 46 25

a.Calculate the standard error.

b.Find t v,a/2(t 分配) for a 95% confidence interval for the true population mean.

### 1 個解答

• 匿名使用者
7 年前
最佳解答

1.

(a)

X=1/2 x1 + 1/2 x2

Y=1/4 x1 + 3/4 x2

Z=1/3 x1 + 2/3 x2

由期望值的線性知道：

E(X)=1/2E(x1+x2)=1/2*2μ=μ

E(Y)=1/4E(x1)+3/4E(x2)=1/4μ+3/4μ=μ

E(Z)=1/3E(x1)+2/3E(x2)=1/3μ+2/3μ=μ

所以X,Y,Z三個點估計式皆為μ之不偏估計式

(b)

因為Var(aX)=a平方Var(x) ,所以

Var(X)=1/4 Var(x1+x2)=1/4*2σ平方=(σ平方)/2

Var(Y)=1/16 Var(x1)+9/16 Var(x2)=(10/16) (σ平方)=5/8(σ平方)

Var(Z)=1/9 Var(x1)+4/9 Var(x2)=(5/9) (σ平方)

又Var(X)<Var(Y)<Var(Z),故X為三者中最有效之估計式

(c)

X相對於Y之效率: [Var(Y)]/[Var(X)]=5/4

X相對於Z之效率: [Var(Z)]/[Var(X)]=10/9