Incoming students at a certain law school have an average...

Incoming students at a certain law school have an average LSAT (Law School Aptitude Test) score of 163 and an $S D$ of 8 . Tomorrow, one of these students will be picked at random. You have to guess the score now; the guess will be compared with the actual score, to see how far off it is. Each point off will cost a dollar. For example, if the guess is 158 and the score is really $151,$ you will have to pay $\$ 7 .$ . (a) Is the best guess $150,163,$ or 170$?$ (b) You have about 1 chance in 3 to lose more than $\longrightarrow$ Fill in the blank. Options: $\$ 1, \$ 8, \$ 20 .$ (LSAT scores range from 120 to 180 ; the average across all test-takers is about 150 and the SD is about $9 .$ The test is re-normed from time to time, the data are for $2005 . )$

Incoming students at a certain law school have an average LSAT (Law School Aptitude Test) score of 163 and an $S D$ of 8 . Tomorrow, one of these students will be picked at random. You have to guess the score now; the guess will be compared with the actual score, to see how far off it is. Each point off will cost a dollar. For example, if the guess is 158 and the score is really $151,$ you will have to pay $\$ 7 .$ .
(a) Is the best guess $150,163,$ or 170$?$
(b) You have about 1 chance in 3 to lose more than $\longrightarrow$ Fill in the blank. Options: $\$ 1, \$ 8, \$ 20 .$
(LSAT scores range from 120 to 180 ; the average across all test-takers is about 150 and the SD is about $9 .$ The test is re-normed from time to time, the data are for $2005 . )$

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Key Concepts

Absolute Loss Minimization

In decision problems where the cost is proportional to the absolute difference between a guess and an outcome, the optimal estimate minimizes the expected absolute error. This principle leads to choosing the median of the probability distribution as the best estimate, because it minimizes the total expected deviations.

Symmetry in Distributions

When dealing with symmetric distributions, such as the normal distribution, the median and the mean coincide. This property implies that if the cost function is based on absolute error, the best guess (the median) will be the same as the average, simplifying the decision process.

Standard Deviation and Probability Thresholds

The standard deviation provides a measure of the spread of the distribution and, in the case of normal distributions, can be used to estimate the probability of outcomes falling a certain distance from the mean. For instance, using the empirical rule, approximately one-third of the outcomes lie beyond one standard deviation, which helps in assessing the likelihood of incurring higher costs.

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