Question

Question 7 Paint viscosity is a measure of the thickness that determines whether the paint will cover in a single coat. A random sample of latex paint viscocities (in Krebs units) was obtained, and the data are given in the following table: 113,124,141,115,115,129,113,129,112,112 a. Find z-score for each observation. b. Find the mean and standard deviation for all of the z-scores c For any set of observations, can you calculate the mean and standard deviation of the corresponding z-scores? Prove your result.

Answer 1

Sample size, n =10

Sample mean of the observations, $\bar{X}$ =$\Sigma Xi/n$ =1203/10 =120.3

Sample std.deviation of the observations, s =$\sqrt{\Sigma (Xi-\bar{X})^2/(n-1)}$ =9.967

a.

Formula: Zi =$(Xi-\bar{X})/s$

For 113, Z1 =(113 - 120.3)/9.967 = -0.732

For 124, Z2 =(124 - 120.3)/9.967 = 0.371

Similarly, the reaming Z scores are: 2.077; -0.532; -0.532; 0.873; -0.732; 0.873; -0.833, -0.833

b.

Mean of Z-scores = $\bar{Z} =\Sigma Z/n$ =0/10 = 0

Std.deviation of Z-scores =s(Z) =$\sqrt{\Sigma (Zi - \bar{Z})^2/(n-1)}$ =1

c.

For any set of observations, we can calculate the mean and std.deviation of the corresponding Z-scores. However, always the mean of the corresponding Z scores is 0 and the std.deviation of the corresponding Z scores is 1.

Here is the proof for the same: