Test the Claim That $\mu _ { 1 } > \mu _ { 2 }$

question 38

Essay

Test the claim that $\mu _ { 1 } > \mu _ { 2 }$ Two samples are random, independent, and come from populations that are
normally distributed. The sample statistics are given below. Assume that $\sigma { } _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 }$ . Use $\alpha$ = 0.01. $$\begin{array} { l l } \mathrm { n } _ { 1 } = 18 & \mathrm { n } _ { 2 } = 13 \\\overline { \mathrm { x } }_ 1 = 520 & \overline { \mathrm { x } _ { 2 } } = 505 \\\mathrm {~s} _ { 1 } = 40 & \mathrm {~s} _ { 2 } = 25\end{array}$$

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Definitions:

Voting Paradox

is a situation in social choice theory where collective preferences can be cyclic (A is preferred to B, B is preferred to C, and C is preferred to A), despite the individual preferences being consistent.

Majority-Rule Voting

A decision-making process where the option that receives more than half of the votes is chosen, commonly used in democratic systems and organizations.

Voting Paradox

A situation in social choice theory where collective preferences can be cyclic (i.e., not transitive), even if the preferences of individual voters are not, leading to a lack of consistent aggregation of individual preferences into a coherent group order.

Impossibility Theorem

A principle, also known as Arrow's impossibility theorem, stating that it is impossible to devise a social welfare function that fairly ranks societal preferences in the presence of three or more options.