Question

1. using the following informal histogram, calculate each of the following.

a. ( n )
b. ( sigma x )
c. ( sigma x^2 )

Explanation:

Part a: Calculating \( n \)

Step 1: Identify frequencies for each \( X \)

For \( X = 1 \), frequency \( f = 1 \); \( X = 2 \), \( f = 3 \); \( X = 3 \), \( f = 3 \); \( X = 4 \), \( f = 4 \); \( X = 5 \), \( f = 2 \); \( X = 6 \), \( f = 1 \).

Step 2: Sum the frequencies

\( n=\sum f = 1 + 3 + 3 + 4 + 2 + 1 \)
\( n = 14 \)

Part b: Calculating \( \sum X \)

Step 1: Multiply each \( X \) by its frequency

• \( X = 1 \): \( 1\times1 = 1 \)
• \( X = 2 \): \( 2\times3 = 6 \)
• \( X = 3 \): \( 3\times3 = 9 \)
• \( X = 4 \): \( 4\times4 = 16 \)
• \( X = 5 \): \( 5\times2 = 10 \)
• \( X = 6 \): \( 6\times1 = 6 \)

Step 2: Sum these products

\( \sum X=1 + 6 + 9 + 16 + 10 + 6 \)
\( \sum X = 48 \)

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

Step 1: Calculate \( X^2 \) for each \( X \) and multiply by frequency

• \( X = 1 \): \( 1^2\times1 = 1\times1 = 1 \)
• \( X = 2 \): \( 2^2\times3 = 4\times3 = 12 \)
• \( X = 3 \): \( 3^2\times3 = 9\times3 = 27 \)
• \( X = 4 \): \( 4^2\times4 = 16\times4 = 64 \)
• \( X = 5 \): \( 5^2\times2 = 25\times2 = 50 \)
• \( X = 6 \): \( 6^2\times1 = 36\times1 = 36 \)

Step 2: Sum these products

\( \sum X^2=1 + 12 + 27 + 64 + 50 + 36 \)
\( \sum X^2 = 190 \)

Answer:

s:
a. \( \boldsymbol{n = 14} \)
b. \( \boldsymbol{\sum X = 48} \)
c. \( \boldsymbol{\sum X^2 = 190} \)