The mean daily rainfall in Los Angeles in December is 0.05 inches with a standard deviation of 0.02 inches. What is the probability that
the total rainfall in Los Angeles for 37 randomly selected December days (possibly over several years) will not exceed 2 inches?
Carry your intermediate computations to at least four decimal places. Report your result to at least three decimal places.

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sum of values of X, i.e ∑X, will be approximately normally distributed.

Then, the mean of the distribution of the sum of values of X is given by,

[tex]\mu_{x}=n\mu[/tex]

And the standard deviation of the distribution of the sum of values of X is given by,

[tex]\sigma_{x}=\sqrt{n}\sigma[/tex]

As the sample size is large, i.e. n = 37 > 30, the Central Limit Theorem can be used to approximate the sampling distribution of sample mean daily rainfall in Los Angeles.

Compute the probability that the total rainfall will not exceed 2 inches as follows:

[tex]P(\sum X\leq 2)=P(\frac{\sum X-n\mu}{\sqrt{n}\sigma}\leq \frac{2-(37\times 0.05)}{\sqrt{37}\times 0.02})\\\\=P(Z<1.23)\\\\=0.89065\\\\\approx 0.8907[/tex]

Thus, the probability that the total rainfall will not exceed 2 inches is 0.8907.