Which of the Following Statements Are True If $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$

question 1

Multiple Choice

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 $\mu \text { and variance } \sigma ^ { 2 }$ ?

• A) \(S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 2 ) \text { is an unbiased estimator of } \sigma ^ { 2 }\)
• B) \(S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n + 1 ) \text { is an unbiased estimator of } \sigma ^ { 2 }\)
• C) \(S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / n \text { is an unbiased estimator of } \sigma ^ { 2 }\)
• D) \(S ^ { 2 } = \sum ( X - \bar { X } ) ^ { 2 } / ( n - 1 ) \text { is an unbiased estimator of } σ^ { 2 }\)
• E) All of the above statements are true provided that the sample size n > 30.

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Definitions:

Neuroticism

A personality trait characterized by emotional instability, anxiety, moodiness, and a high degree of stress response.

Agreeableness

A personality trait characterized by attributes such as trust, altruism, kindness, and affection towards others.

Openness

The willingness to consider new ideas, experiences, and perspectives.

Self-Concept

A unifying construct of personality; the most important determinant of human behavior; the basis of present identity and future conduct; the primary factor in character formation.