Please explain to me why you are square rooting the pooled variance, divided by the sample size plus the other sample size, and then dividing again by the confidence interval when you clearly did not teach that. That is not how it shows to calculate the confidence interval of two independent variables with variances unknown and assumed to be equal. NO WHERE does it say to use the margin of error to calculate it and no where does it say to sqrt the pooled variance, divided by the sample size plus the other sample size, and then dividing again by the confidence interval.
This is not the first time I have come across a situation where your lecture has said one thing and then in examples you clearly did something against what you have said. This is beyond frustrating when trying to learn something. If you are going to change the equation completely when calculating it then YOU NEED TO STATE that in the lecture. That’s poor teaching!