default rate 为什么是sp和fitch的均值呢？

问题如下：

Suppose you wish to test whether the default rate of bonds that are rated as investment grade by S&P is the same as the default rate on bonds rated as investment grade by Fitch. What are the null and alternative hypotheses? What data would you need to test the null hypothesis?

解释：

The null is that the IG (investment grade) default rate is the same for S&P rated firms as it is for Fitch-rated firms. If Si are default indicators for S&P rated firms, and Fi are default indicators for Fitch-rated firms, then the null is $H_0: \mu_S = \mu_F$

which states that the mean values are the same.

The null can be equivalently expressed as

$H_0: \mu_S - \mu_F = 0$

The alternative is

$H_1: \mu_S \neq \mu_F$

or equivalently

$H_1: \mu_S - \mu_F \neq 0$

The data required to test this hypothesis would be binary random variables where 1 indicates that an IG bond defaulted and 0 if it did not within a fixed time frame (e.g., a quarter).