A Particular Brand of Dishwasher Soap Is Sold in Three $X _ { 1 } \text { and } X _ { 2 }$

question 17

Essay

A particular brand of dishwasher soap is sold in three sizes: 25oz, 40oz, and 65 oz. Twenty percent of all purchasers select a 25 oz box, fifty percent select a 40 oz box, and the remaining thirty percent choose a 65 oz box. Let $X _ { 1 } \text { and } X _ { 2 }$ denote the package sizes selected by two independently selected purchasers.
a. Determine the sampling distribution of $\bar { X }$
, calculate $E ( \bar { X } )$
, and compare to $\mu .$
b. Determine the sampling distribution of the sample variance $S ^ { 2 } \text {, calculate } E \left( S ^ { 2 } \right) \text {, and compare to } \sigma ^ { 2 } \text {. }$

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Definitions:

Hypnosis

A trance-like state of focused attention and increased suggestibility, often used therapeutically to modify behaviors and alleviate symptoms.

Relaxation Training

Techniques aimed at reducing stress and tension, including exercises like deep breathing, meditation, or progressive muscle relaxation.

Mindfulness Meditation

A practice of focused attention and awareness on the present moment, often used as a technique to manage stress, reduce anxiety, and promote emotional health.

Temporary Assistance

Governmental programs designed to provide financial aid or support to individuals or families in need for a limited period of time.