The reason speaking with conviction is important is because it is a way of conveying confidence as a speaker. I feel it is important to sound confident in yourself because that will allow other people to listen, or trust what you are saying.
For example, I will give two sentences with the same meaning, and let you decide which has more conviction to it... Jack said, "Hey it is going to rain today so you should bring an umbrella to school with you." While Jill said, "Hey I do not know for sure if it will rain or not, but it might." So as you can see in these two sentences, Jack has more conviction then Jill does. Therefore, it is easier to believe Jack than Jill.
As for applying this subject to class work, I think I can work on speaking with conviction in every one of our assignments. Every assignment I complete from here on out, I will keep "conviction" in the back of my mind. Especially in my future blog posts, I will convey confidence in my own personal self. I hope pursuing coursework in this field will allow me to practice everything that I have already talked about, which is now speaking with conviction.