Explanation:
The z-statistic equals:
where is the value for a randomly selected observation from the population, is the mean value for the population, and is the standard deviation of the population. Therefore, as indicated by the formula, the z-statistic is the number of standard deviations is from the mean (The first analyst is correct).
According to the normal distribution, 95% of the observations lie within 1.96 standard deviations of the mean, which implies that 95% of the z-statistics lie within plus and minus 1.96. Therefore, 5% of the z-statistics lie above plus 1.96 and below minus 1.96 and since the normal distribution is symmetrical, then 2.5% of the z-statistics lie below minus 1.96. As a result, 97.5% (not 95%) of the z-statistics lie above minus 1.96. (The second analyst is not correct).
(Book 2, Module 14.2, LO 14.a)
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Question 27
Two risk analysts recently discussed the application of the normal distribution for random variables. The first analyst claimed that the z-statistic measures the distance, in standard deviation units, that a given observation is from the population mean. The second analyst claimed that there is a 95% chance that the z-statistic lies above negative 1.96. Regarding the statements of the analysts:
A
the first analyst is correct; the second analyst is correct.
B
the first analyst is correct; the second analyst is incorrect.
C
the first analyst is incorrect; the second analyst is correct.
D
the first analyst is incorrect; the second analyst is incorrect.
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