If I want to test two hypothesis, i.e. whether Treatment 1 (T1) affects outcome O, and whether Treatment (T2) affects outcome O, relative to a control group (C), does it make a difference with respect to the Type I error by, whether I

a) have one control group for each treatment group, i.e. compare means of outcomes in C1 = T1, and C2 = T2

b) have only one control group, i.e. compare means of outcomes in C1 = T1, and C1 = T2,

each with a two-sample t-test (there should be no significant difference of outcomes between C1 and C2).

Does b) inflate the Type I error because I use C1 for both tests?

Thanks in advance.

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