Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$

question 44

Multiple Choice

Let $X \square \operatorname { Bin } \left( m , p _ { 1 } \right) \text { and } Y \square \operatorname { Bin } \left( n , p _ { 2 } \right)$ with X and Y independent variables, and let $\hat { p } _ { 1 } = X / m \text { and } \hat { p } _ { 2 } = Y/ n$ Which of the following statements are not correct?

• A) \(E \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } - p _ { 2 } \text {, so } \hat { p } _ { 1 } - \hat { p } _ { 2 }\)
• Is an unbiased estimator of \(p _ { 1 } - p _ { 2 }\)
• B) When both m and n are large, the estimator \(\hat { p } _ { 1 } - \hat { p } _ { \mathbf { 2 } }\)
• Individually has approximately normal distributions.
• C) When both m and n are large, the estimator \(\hat { p } _ { 1 } - \hat { p } _ { 2 }\)
• Has approximately a normal distribution.
• D) \(V \left( \hat { p } _ { 1 } - \hat { p } _ { 2 } \right) = p _ { 1 } q _ { 1 } / m \quad p _ { 2 } q _ { 2 } / n , \text { where } q _ { l } = 1 - p _ { l } \text { for } i = 1,2\)
• E) All of the above statements are correct.

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Definitions:

Volunteer Bias

The distortion of research results due to differences between those who choose to participate in studies and those who do not.

Representative Sample

A portion of the population that correctly mirrors the characteristics of the whole population.

Operationally Defined

A term or variable is defined by the specific process or set of criteria by which it is measured or determined in a study or experiment.

Participant Observation

A research method where the investigator actively engages in the community or situation being studied while observing its dynamics.