Question

The national weather service keeps records of rainfall in valleys. Records indicate that in a certain valley the annual rainfall has a mean of 95 inches and a standard deviation of 12 inches. Suppose the rainfalls are sampled during randomly picked years and x is the mean amount of rain in these years. For samples of size 36 determine the mean and standard deviation of x.

Answer 1

Answer:

The mean is 95 and the standard deviation is 2

Step-by-step explanation:

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question:

Population:

Mean 95, Standard deviation 12

Samples of size 36:

By the Central Limit Theorem,

Mean 95

Standard deviation $s = \frac{12}{\sqrt{36}} = 2$