Question

A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to:

• A. \(5.96 \)
• B. \(6.14 \)
• C. \(6.04 \)
• D. \(6.24 \)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We have the statistical parameters (count, mean, variance) of two distinct groups. We need to merge these groups and find the variance of the combined dataset.
Step 2: Key Formula or Approach:
The combined mean of two groups is given by:
\( \bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2} \)
The combined variance formula is:
\( \sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2} \)
where \( d_1 = \bar{x}_1 - \bar{x} \) and \( d_2 = \bar{x}_2 - \bar{x} \).
Step 3: Detailed Explanation:
Given data:
Set 1: \( n_1 = 4 \), \( \bar{x}_1 = 1 \), \( \sigma_1^2 = 13 \)
Set 2: \( n_2 = 6 \), \( \bar{x}_2 = 2 \), \( \sigma_2^2 = 1 \)
First, calculate the combined mean \( \bar{x} \):
\( \bar{x} = \frac{4(1) + 6(2)}{4 + 6} = \frac{4 + 12}{10} = \frac{16}{10} = 1.6 \).
Next, calculate the deviations \( d_1 \) and \( d_2 \) from the combined mean:
\( d_1 = \bar{x}_1 - \bar{x} = 1 - 1.6 = -0.6 \implies d_1^2 = 0.36 \)
\( d_2 = \bar{x}_2 - \bar{x} = 2 - 1.6 = 0.4 \implies d_2^2 = 0.16 \)
Now, substitute all values into the combined variance formula:
\( \sigma^2 = \frac{4(13 + 0.36) + 6(1 + 0.16)}{10} \)
\( \sigma^2 = \frac{4(13.36) + 6(1.16)}{10} \)
Calculate the numerator:
\( 4 \times 13.36 = 53.44 \)
\( 6 \times 1.16 = 6.96 \)
Total numerator = \( 53.44 + 6.96 = 60.40 \).
Now divide by the total number of observations (10):
\( \sigma^2 = \frac{60.40}{10} = 6.04 \).
Step 4: Final Answer:
The variance of all these 10 observations is 6.04.