Let’s learn more about Type I error.

Warning, Warning, Danger, Young Statistician!
If we calculate all t tests, we will dramatically increase the type 1 error rate!
Type 1 error is saying there is a difference when there isn't one! There will be a 26.5% chance that we make at least 1 type error! I's like Russian Roulette, there is a one in 6 chance of disaster each time you play, but if you play 6 times, the odds you shoot yourself at least once is 26.5%.

The F-ratio avoids this problem because it is an overall test.