Question

b. $\sigma x$
c. $\sigma x^3$

1. find each of the following values for the distribution

shown in the following polygon.

a. $n$
b. $\sigma x$
c. $\sigma x^2$

Explanation:

Part a: Find \( n \)

Step1: Identify frequencies for each \( X \)

From the polygon, the frequency \( f \) for each \( X \) is:

• \( X = 0 \): \( f = 0 \)
• \( X = 1 \): \( f = 1 \)
• \( X = 2 \): \( f = 3 \)
• \( X = 3 \): \( f = 6 \)
• \( X = 4 \): \( f = 5 \)
• \( X = 5 \): \( f = 2 \)
• \( X = 6 \): \( f = 0 \)

Step2: Sum the frequencies (\( n = \sum f \))

\( n = 0 + 1 + 3 + 6 + 5 + 2 + 0 = 17 \)

Part b: Find \( \sum X \)

Step1: Calculate \( X \cdot f \) for each \( X \)

• \( X = 0 \): \( 0 \cdot 0 = 0 \)
• \( X = 1 \): \( 1 \cdot 1 = 1 \)
• \( X = 2 \): \( 2 \cdot 3 = 6 \)
• \( X = 3 \): \( 3 \cdot 6 = 18 \)
• \( X = 4 \): \( 4 \cdot 5 = 20 \)
• \( X = 5 \): \( 5 \cdot 2 = 10 \)
• \( X = 6 \): \( 6 \cdot 0 = 0 \)

Step2: Sum the \( X \cdot f \) values

\( \sum X = 0 + 1 + 6 + 18 + 20 + 10 + 0 = 55 \)

Part c: Find \( \sum X^2 \)

Step1: Calculate \( X^2 \cdot f \) for each \( X \)

• \( X = 0 \): \( 0^2 \cdot 0 = 0 \)
• \( X = 1 \): \( 1^2 \cdot 1 = 1 \)
• \( X = 2 \): \( 2^2 \cdot 3 = 4 \cdot 3 = 12 \)
• \( X = 3 \): \( 3^2 \cdot 6 = 9 \cdot 6 = 54 \)
• \( X = 4 \): \( 4^2 \cdot 5 = 16 \cdot 5 = 80 \)
• \( X = 5 \): \( 5^2 \cdot 2 = 25 \cdot 2 = 50 \)
• \( X = 6 \): \( 6^2 \cdot 0 = 0 \)

Step2: Sum the \( X^2 \cdot f \) values

\( \sum X^2 = 0 + 1 + 12 + 54 + 80 + 50 + 0 = 197 \)

Answer:

s:
a. \( \boldsymbol{n = 17} \)
b. \( \boldsymbol{\sum X = 55} \)
c. \( \boldsymbol{\sum X^2 = 197} \)