Approximate with a Normal distribution (Bayesian statistics)

Dear users,

I am solving an exercise.

Here, assume a Gamma distribution with a parameter lambda I have a priori:

Pr(lambda is (belongs) [10,20])=0.95

I have to approximate the priori with a Normal distribution.

When I saw the solution, it says:

after approximation with a normal distribution, we obtain the confidence interval: lambda belongs [15+-5]

Which results in:

E[lambda]=15

V[lambda]=(5/1.96)^2=6.5.

Please, could you explain me, how to approximate with a Normal distribution. And could you give me the formula itself. I cannot understand how to get the confidence interval [15+-5].

What about the mean and the variance, I understood.

Thank you in advance, I highly appreciate.