Poisson Approximation to Normal Distribution

I understand that for large $$\lambda$$ we can approximate Poisson to normal distribution. But I am not sure if we should approximate to this:
N($$\lambda$$,$$\lambda$$) or N($$\mu$$,$$\sigma^2$$)

I am trying to solve this question. But I know I can't use Poisson because the mean is not equal to the variance.

$$\mu=30=\sigma^2$$ but... $$\sigma=\sqrt{λ}=\sqrt{30}\ne3$$

It is known that, on an average day, 30 patients will see this doctor, with a standard deviation of 3 patients per day.

a)Describe a model you might use to model the number of patients on a given day .

b) On one particular day, 45 patients visit the doctor. Considering the model you developed in your answer to the previous question, do you think that this number of patients in a given day is cause for alarm? Use calculations to back up your answer by determining the probability of seeing 45 or more patients in a given day.

Any help or clarification would be helpful. Thanks

A useful tool to measure if your data is normally distributed is the coefficient of variation, which is : $$\frac{standard deviation}{average}$$ This thus uses the first two moments and in practice, if this ratio stays below $$0.5$$, you can assume a normal distribution. Which is the case in your question. Just use the normal distribution and not the poisson distribution. Because negative patients can't happen, maybe you can also go for the gamma distribution.