Exam 6: Point Estimation

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 __________.

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 $\mu$ ?

Which of the following statements are not always true?

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.

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

Which of the following statements are not true?

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 __________.

Which of the following statements are correct?