Quiz 4 PSTAT 120B

1.设Y1,。 。 。 ,Yn〜Bernoulli(θ)。在测验3中,您得出了一个大样本置信区间

θ;在这里,您将开发假设􏰈H0的大样本低尾假设检验:θ=θ0

HA:θ<θ0

使用决策规则T(Y)∈R其中
√n(Ȳ-θ0)

R =(-∞,c)
−→N(0,1)。使用此结果可写出大约

T(Y)=􏰉Ȳ(1-Ȳ)和

√n(Ȳ−θ)

(a)根据MLE属性√̄

Y(1-Y)

CDF。

(b)通过求解c的方程Q(θ0)=α,找到给出水平α测试的c的值。

(c)根据标准高斯CDF推导用于计算检验的p值的表达式。

d
以标准高斯为单位的幂函数Q(θ)=Pθ(T(Y)∈R)的形式

2个

测验4 PSTAT 120B

2.下表显示了2011年3月当前人口调查的工资数据摘要统计。数据是美国1000位男性调查受访者的子样本;该表显示了没有大学学位,大学或高级学位的黑人和白人受访者的团体平均工资和标准差,以及团体人数。在这里,您将研究教育和种族对收入的影响。

种族黑色黑色

以下问题。一定要制定
用文字和符号表示,说明测试统计量和拒绝区域,计算p值,

并以α= 0.05的水平解释测试的结论。

(a)高等教育与黑人收入增加有关吗? (b)高等教育与白人男子的收入增加相关吗?

(c)是否有证据表明受过高等教育的男子存在种族收入差距? (d)是否有证据表明没有学位的人的种族收入差距?

白色白色

教育没有大专​​院校或没有大专院校或

级别更高更高

每周收入(平均)636.35
1245.13
855.23

每周收入(标准差)362.50 687.15 522.68 760.16

n 53 21588338

1523.45进行假设检验以评估

1. Let Y1, . . . , Yn Bernoulli(θ). In Quiz 3, you derived a large-sample confidence interval for

θ; here you will develop a large-sample lower-tailed hypothesis test of the hypotheses 􏰈H0: θ=θ0

HA: θ<θ0

using the decision rule T (Y ) R where
n ( Y ̄ θ 0 )

R=(−∞,c)
−→ N (0, 1). Use this result to write an approx-

T(Y)= 􏰉Y ̄(1Y ̄) and

n ( Y ̄ θ )

  1. (a)  Based on the MLE properties, ̄ ̄

    Y (1Y )

    CDF.

  2. (b)  Find the value of c that gives a level α test by solving the equation Q(θ0) = α for c.
  3. (c)  Derive an expression for computing the p-value for the test in terms of the standard Gaussian CDF.

d
imation of the power function Q(θ) = Pθ (T (Y ) R) in terms of the standard Gaussian

2

Quiz 4 PSTAT 120B

2. The table below shows summary statistics for wage data from the March 2011 Current Pop- ulation Survey. The data are a subsample of 1000 male survey respondents in the U.S.; the table shows group average wages and standard deviations for black and white respondents with no college degree and a college or advanced degree, along with the group sizes. Here you will investigate the effect of education and race on income.

Race Black Black

following questions. Be sure to formulate
ses in words and symbols, state the test statistic and rejection region, compute the
p-value,

and interpret the conclusion of the test at level α = 0.05.

(a) Is higher education associated with an increase in income for black men? (b) Is higher education associated with an increase in income for white men?

(c) Is there evidence of a racial income disparity for highly educated men? (d) Is there evidence of a racial income disparity for men without degrees?

White White

Education No college College or No college College or

Level higher higher

Weekly Earnings (mean) 636.35
1245.13
855.23

Weekly Earnings (sd) 362.50 687.15 522.68 760.16

n 53 21 588 338

1523.45 Conduct hypothesis tests to assess the

the hypothe-

3


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