The Data for a Random Sample of Six Paired Observations

question 5

Essay

The data for a random sample of six paired observations are shown below. $\begin{array} { l c c } \text { Pair } & \text { Observation 1 } & \text { Observation 2 } \\ 1 & 1 & 3 \\ 2 & 2 & 4 \\ 3 & 3 & 5 \\ 4 & 4 & 6 \\ 5 & 5 & 7 \\ 6 & 6 & 8 \end{array}$

a. Calculate the difference between each pair of observations by subtracting observation 2 from observation 1 . Use the differences to calculate $s ^ { 2 }$ .
b. Calculate the standard deviations $s _ { 1 } { } ^ { 2 }$ and $s _ { 2 } { } ^ { 2 }$ of each column of observations. Then find pooled estimate of the variance $s p ^ { 2 }$ .
c. Comparing $s _ { d } { } ^ { 2 }$ and $s p ^ { 2 }$ , explain the benefit of a paired difference experiment.

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Definitions:

Place Theory

A theory in the field of psychophysics that suggests the perception of sound depends on where each component frequency produces vibrations along the basilar membrane of the inner ear.

Frequency Theory

A theory related to hearing that suggests the brain detects the frequency of an auditory signal by the rate at which the auditory nerve fires.

Taste Sensations

The experience or perception of taste, typically categorized into basic tastes such as sweet, sour, salty, bitter, and umami.

Salt

A mineral composed primarily of sodium chloride (NaCl), used in cooking and preserving food, and is essential for human health in small amounts.