You can do the t-test if the mortality ratio in BOTH types of hospitals (with staffing problems / without staffing problems) is normal, and if the variance of the ratio in the two types are equal.

Since your calculated sample standard deviations are pretty close, I'd say the equal variance assumption is okay.

A non-parametric alternative to the t-test is the Mann-Whitney test.

Note that the mean for the first column of data is 1.09; I input your data to Excel and got t=1.62. So you might have a calculation error.

You can also use Excel to get a more precise p-value:=TDIST(1.46,11,2) where the first # is your test statistic, second # is your df, third # is 1 (one-tail test) or 2 (two-tail test)