Question

Each of the following samples was obtained from a population with µ = 100 and σ = 10. Find
the z-score corresponding to each sample mean.
a. M = 95 for a sample of n = 4
b. M = 104 for a sample of n = 25
c. M = 103 for a sample of n = 100

Answer 1

If X has a distribution with mean $\mu,$ and standard deviation $\sigma,$ and is approximately normally distributed or $n$ is large, then the sample mean M is approximately normally distributed with mean $\mu$ and standard error $\sigma/\sqrt{n}.$

$$z={M-\mu \over \sigma/\sqrt{n}}$$

Given that $\mu=100$ and $\sigma=10.$

a. M = 95 for a sample of n = 4 

$$z={95-100 \over 10/\sqrt{4}}=-1$$

b. M = 104 for a sample of n = 25 

$$z={104-100 \over 10/\sqrt{25}}=2$$

c. M = 103 for a sample of n = 100

$$z={103-100 \over 10/\sqrt{100}}=3$$