Let $X$ Denote the Proportion of Allotted Time That a Randomly Selected

question 24

Essay

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.

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Definitions:

Eco-Map

A graphical representation that shows all of the systems at play in an individual's life, often used in social work and family therapy.

Social Environment

The immediate physical and social setting in which people live or in which something happens or develops, including the culture, institutions, and individuals they interact with.

Client Family

A term used in social work and counseling to refer to the group of individuals related by blood, marriage, or other significant relational bonds, who are the focus of therapeutic or social support interventions.

Eco-Map

A graphical representation that shows an individual's or family's relationships with their social and physical environment.