NO.PZ2023091601000063
问题如下：
A senior manager on the proprietary trading
desk of an investment bank is evaluating the performance of two fixed-income
traders, trader A and trader B, using their annual performance over the last 10
years. Trader A generated an average return of 7% with a standard deviation of
15%, while trader B generated an average return of 12% with a standard
deviation of 20%. The manager tests the null hypothesis that the traders
performed equally well against the alternative hypothesis that the average
return of trader B is higher than the average return of trader A. Assuming the
performances of each trader are independent, which of the following correctly identifies
the test statistic and the 5% critical value corresponding to this alternative
hypothesis?
选项：
A.The test statistic
is -0.63 and the critical value is 2.31.
The test statistic
is -0.27 and the critical value is 2.31.
The test statistic
is 0.27 and the critical value is 1.86.
The test statistic
is 0.63 and the critical value is 1.86.
解释：
D is correct. If
we let X and Y denote traders B and A respectively (noting that it is safest to
work with proportions rather than percentages) the test statistic for testing
that the means are equal (i.e., 𝐻0:𝜇𝑋=𝜇𝑌) is:
Note
that the denominator no longer has the (2σXY/n) component in the calculation;
this is because the two performance results are independent – as such, the
correlation between X and Y is 0. The statistic 0.63 follows a t-distribution
with 8 degrees of freedom and a 5% (one-sided) size of test uses the critical
value of 1.86.
题目中“ the null hypothesis that the traders performed equally well”原假设显示此题是双尾，5%显著性水平，因此我理解关键值应该对应2.5% one side? 还有分子是12%-7%，为什们不是7%-12%？
