Question 1

Which of the following statements are not always true?

Accepted Answer

A)   A point estimator  θ^\hat { 	heta }θ^ Is said to be an unbiased estimator of parameter  θ	hetaθ If  E(θ^) =θE ( \hat { 	heta } )  = 	hetaE(θ^) =θ For every possible value of  θ	hetaθ ) 
B)   If the estimator  θ^\hat { 	heta }θ^ Is not unbiased of parameter  θ	hetaθ , the difference  E(θ^) −θE ( \hat { 	heta } )  - 	hetaE(θ^) −θ Is called the bias of  θ^\hat { 	heta }θ^ ) 
C)   A point estimator  θ^\hat { 	heta }θ^ Is unbiased if its probability sampling distribution is always "centered" at the true value of the parameter  θ	hetaθ , where "centered" here means that the median of the distribution of  θ^ is θ\hat { 	heta } 	ext { is } 	hetaθ^ is θ ) 
D)   All of the above statements are true.

Question 2

Which of the following statements are not true?

Accepted Answer

A)   The symbol  θ^\hat { 	heta }θ^ Is customarily used to denote the estimator of parameter  θ	hetaθ And the point estimate resulting from a given sample.
B)   The equality  μ^=Xˉ\hat { \mu } = \bar { X }μ^​=Xˉ Is read as "the point estimator of  Xˉ is μ^."\bar { X } 	ext { is } \hat { \mu } . "Xˉ is μ^​." 
C)   The difference between  θ^\hat { 	heta }θ^ And the parameter  θ	hetaθ Is referred to as error of estimation.
D)   None of the above statements is true.

Question 3

Which of the following statements are true if  X1,X2,⋯⋯ ,XnX _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }X1​,X2​,⋯⋯,Xn​  is a random sample from a distribution with mean  μ\muμ  ?

Accepted Answer

A)   The sample mean  Xˉ\bar { X }Xˉ Is always an unbiased estimator of  μ\muμ ) 
B)   The sample mean  μ~	ilde { \mu }μ~​ Is an unbiased estimator of  μ\muμ If the distribution is continuous and symmetric.
C)   Any trimmed mean is an unbiased estimator of  μ\muμ If the distribution is continuous and symmetric.
D)   None of the above statements are true.
E)   All of the above statements are true.

Question 4

Which of the following statements are correct?

Accepted Answer

A)   The first population moment is  μ\muμ , while the first sample moment is  Xˉ\bar { X }Xˉ ) 
B)   The moment estimators  θ^1,⋯⋯ ,θ^m\hat { 	heta } _ { 1 } , \cdots \cdots , \hat { 	heta } _ { m }θ^1​,⋯⋯,θ^m​ Are obtained by equating the first m sample moments to the corresponding first m population moments, and solving for the unknown parameters  θ1,⋯⋯ ,θm	heta _ { 1 } , \cdots \cdots , 	heta _ { m }θ1​,⋯⋯,θm​ ) 
C)   The method of maximum likelihood was first introduced by R.A. Fisher, a geneticist and statistician, in the 1920's.
D)   All of the above statements are true.
E)   Only (
A)   and (
B)   are true.

Question 5

The sample median $\tilde { \mu }$ and any trimmed mean are unbiased estimators of the population mean $\mu$ if the random sample from a population that is __________ and __________.

Accepted Answer

continuous...

Question 6

In your text, two important methods were discussed for obtaining point estimates: the method of __________ and the method of __________.

Accepted Answer

moments, m...

Question 7

Let $X$ denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of $X$ is $f ( x ; \theta ) = \left\{ \begin{array} { l l } 
( \theta + 1 ) x ^ { \theta } & 0 \leq x \leq 1 \
0 & \text { otherwise }
\end{array} \right.$ where $\theta$ > -1. A random sample of ten students yields data $x _ { 1 } = .92 , x _ { 2 } = .79 , x _ { 3 } = .90 , x _ { 4 } = .65 , x _ { 5 } = .86 , x _ { 6 } = .47 , x _ { 7 } = .73 , x _ { 8 } = .97 , x _ { 9 } = .94 , x _ { 10 } = .77$ 
a. Use the method of moments to obtain an estimator of $\theta$ 
and then compute the estimate for this data. 
b. Obtain the maximum likelihood estimator of $\theta$ 
and then compute the estimate for the given data.

Accepted Answer

a.

so the moment estimator

i...

Question 8

Let $X _ { 1 } , \cdots \cdots , X _ { n }$ be a random sample from a probability mass function or probability density function f(x). For k = 1,2,3,……, the kth population moment is denoted by __________, while the kth sample moment is __________.

Accepted Answer

The answer of Let \(X _ { 1 } ,...

Exam 6: Point Estimation

Which of the following statements are not always true?

Correct Answer
verified

Let $X _ { 1 } , \cdots \cdots , X _ { n }$ be a random sample from a probability mass function or probability density function f(x). For k = 1,2,3,……, the kth population moment is denoted by , while the kth sample moment is .

Correct Answer
verified

Correct Answer
verified
a. so the moment estimator i...
View Answer

Which of the following statements are not true?

Correct Answer
verified

The sample median $\tilde { \mu }$ and any trimmed mean are unbiased estimators of the population mean $\mu$ if the random sample from a population that is and .

Correct Answer
verified
continuous...
View Answer

Which of the following statements are true if $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$ is a random sample from a distribution with mean $\mu$ ?

Correct Answer
verified

In your text, two important methods were discussed for obtaining point estimates: the method of and the method of .

Correct Answer
verified
moments, m...
View Answer

Which of the following statements are correct?

Correct Answer
verified

Exam 6: Point Estimation

Which of the following statements are not always true?

Correct AnswerverifiedShow Answer

Let X1,⋯⋯ ,XnX _ { 1 } , \cdots \cdots , X _ { n }X1​,⋯⋯,Xn​ be a random sample from a probability mass function or probability density function f(x). For k = 1,2,3,……, the kth population moment is denoted by __________, while the kth sample moment is __________.

Correct AnswerverifiedShow Answer

Correct Answerverifieda. so the moment estimator i...View AnswerShow Answer

Which of the following statements are not true?

Correct AnswerverifiedShow Answer

The sample median μ~\tilde { \mu }μ~​ and any trimmed mean are unbiased estimators of the population mean μ\muμ if the random sample from a population that is __________ and __________.

Correct Answerverifiedcontinuous...View AnswerShow Answer

Which of the following statements are true if X1,X2,⋯⋯ ,XnX _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }X1​,X2​,⋯⋯,Xn​ is a random sample from a distribution with mean μ\muμ ?

Correct AnswerverifiedShow Answer

In your text, two important methods were discussed for obtaining point estimates: the method of __________ and the method of __________.

Correct Answerverifiedmoments, m...View AnswerShow Answer

Which of the following statements are correct?

Correct AnswerverifiedShow Answer

Correct Answer
verified

Let $X _ { 1 } , \cdots \cdots , X _ { n }$ be a random sample from a probability mass function or probability density function f(x). For k = 1,2,3,……, the kth population moment is denoted by , while the kth sample moment is .

Correct Answer
verified

Correct Answer
verified
a. so the moment estimator i...
View Answer

Correct Answer
verified

The sample median $\tilde { \mu }$ and any trimmed mean are unbiased estimators of the population mean $\mu$ if the random sample from a population that is and .

Correct Answer
verified
continuous...
View Answer

Which of the following statements are true if $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$ is a random sample from a distribution with mean $\mu$ ?

Correct Answer
verified

In your text, two important methods were discussed for obtaining point estimates: the method of and the method of .

Correct Answer
verified
moments, m...
View Answer

Correct Answer
verified