Infer confidence intervals of MLE using fisher information

#1
I have some count data that looks to be Poisson. After a goodness of fit, chi-squared test with p=0.5, I am comfortable saying it is from a Poisson distribution. I want to get confidence intervals for the mean/lambda. Can I use fishers information to find the variance of the MLE, and then use that to infer confidence intervals?

Example; Lambda.hat = 3, n =150, and Fisher(lambda.hat) = 50, thus var(lambda.hat) = 0.02
95% confidence intervals = lambda.hat +- 1.96 * sqrt(Var(lambda.hat)) = 3 +- .277 = (2.722, 3.277)

I guess what I'm asking is, can I use the variance from the fisher method instead of the sample variance to infer confidence intervals?