题目
Colleagues Ariel and Bob recently discussed the application of the normal distribution for random variables. Ariel claimed that the z-statistic measures the distance, in standard deviation units, that a given observation is from the population mean. Bob claimed that there is a 95% chance that the z-statistic lays above negative 1.96. Regarding the statements of Ariel and Bob:
选项
A.Ariel is correct; Bob is correct.
B.Ariel is correct; Bob is incorrect.
C.Ariel is incorrect; Bob is correct.
D.Ariel is incorrect; Bob is incorrect.
答案
B
解析
The z-statistic equals (x-μ)/σ, where x is the value for a randomly selected observation from the population, μ is the mean value for the population, and σ is the standard deviation of the population. Therefore, as indicated by the formula, the z-statistic is the number of standard deviations x is from the mean. (Ariel 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. (Bob is not correct). z统计量等于（x-μ）/σ，其中x是从总体中随机选择的观察值，μ是总体的平均值，σ是总体的标准偏差。 因此，如公式所示，z统计量是标准偏差的数量x与平均值的差。 （Ariel是对的）。根据正态分布，95％的观测值在平均值的1.96标准偏差之内，这意味着95％的z统计量在正负1.96范围之内。 因此，z统计量的5％位于正值1.96之上和负值1.96之下，并且由于正态分布是对称的，因此2.5％的z统计量位于负值1.96以下。 结果，z统计量的97.5％（不是95％）位于负1.96之上。 （Bob不正确）。