The lengths of adult males' hands are normally distributed with a mean of 190 mm and the standard deviation is 7.4 mm. Suppose that 44 individuals are randomly chosen. Round all answers to 4 where possible. a. What is the distribution of ¯ x ? ¯ x ~ N b. For the group of 44, find the probability that the average hand length is more than 189.  c. Find the first quartile for the average adult male hand length for this sample size.

Sampling Distribution of a Normal Variable : 

Given a random variable X. Suppose that the population distribution of X is known to be normal, with mean µ and
variance σ² , that is, X ~ N(µ, σ). Then, for any sample size n, it follows that the sampling distribution of $\overline{X}$ is normal,
with mean µ and variance σ²/n , that is,  

                       $\overline{X} ~ N\left(\mu ,\frac{\sigma }{\sqrt{n}}\right)$

• $\frac{\sigma }{\sqrt{n}}$ is called the “standard error of the mean,” denoted SEM, or more simply, s.e. 

The corresponding Z-score transformation formula is $Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}} ~ N(0,1)$