Question

1. for the following set of scores, find the value of each expression: a. σ(x - 4)² b. (σx)² c. σx² d. σ(x + 3)

x
-1
-3
6
-4
0

Explanation:

Step1: Identify the scores

The scores are \(x_1=-1,x_2 = - 3,x_3=6,x_4=-4,x_5 = 0\).

Step2: Calculate \(\sum(X - 4)^2\)

First, find \(x_i-4\) for each \(i\):
\(x_1 - 4=-1 - 4=-5\), \((x_1 - 4)^2=(-5)^2 = 25\);
\(x_2-4=-3 - 4=-7\), \((x_2 - 4)^2=(-7)^2 = 49\);
\(x_3-4=6 - 4 = 2\), \((x_3 - 4)^2=2^2=4\);
\(x_4-4=-4 - 4=-8\), \((x_4 - 4)^2=(-8)^2 = 64\);
\(x_5-4=0 - 4=-4\), \((x_5 - 4)^2=(-4)^2 = 16\).
Then \(\sum(X - 4)^2=25 + 49+4+64+16=158\).

Step3: Calculate \((\sum X)^2\)

\(\sum X=-1-3 + 6-4+0=-2\), so \((\sum X)^2=(-2)^2 = 4\).

Step4: Calculate \(\sum X^2\)

\(x_1^2=(-1)^2 = 1\), \(x_2^2=(-3)^2 = 9\), \(x_3^2=6^2 = 36\), \(x_4^2=(-4)^2 = 16\), \(x_5^2=0^2 = 0\).
\(\sum X^2=1+9+36+16+0=62\).

Step5: Calculate \(\sum(X + 3)\)

\(x_1+3=-1 + 3=2\), \(x_2+3=-3 + 3=0\), \(x_3+3=6 + 3=9\), \(x_4+3=-4 + 3=-1\), \(x_5+3=0 + 3=3\).
\(\sum(X + 3)=2+0+9-1+3=13\).

Answer:

a. \(158\)
b. \(4\)
c. \(62\)
d. \(13\)