© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Given Information
We have 7 observations in total. Their mean is 8 and their variance is 16. Among these 7 observations, two particular values are known: 6 and 8. We need to find the variance of the remaining 5 observations.
Step 2: Express the Mean Condition
The mean of the 7 observations is 8. Hence, the sum of the 7 observations is:
$7 \times 8 = 56$.
Since two of the observations are 6 and 8, let the remaining five observations be $x_1, x_2, x_3, x_4, x_5$. Therefore,
$$
x_1 + x_2 + x_3 + x_4 + x_5 + 6 + 8 = 56.
$$
From this, we get:
$$
x_1 + x_2 + x_3 + x_4 + x_5 = 56 - 14 = 42.
$$
Step 3: Express the Variance Condition
The variance of the 7 observations is 16. By definition:
$$
\text{Variance} = \frac{\sum (\text{observations})^2}{7} - (\text{mean})^2.
$$
So,
$$
16 = \frac{\sum (\text{observations})^2}{7} - 8^2.
$$
Simplifying,
$$
\frac{\sum (\text{observations})^2}{7} = 16 + 64 = 80.
$$
Therefore,
$$
\sum (\text{observations})^2 = 80 \times 7 = 560.
$$
Step 4: Find the Sum of Squares of the Remaining 5 Observations
We know two of the observations are 6 and 8. Their squares are:
$$
6^2 = 36 \quad \text{and} \quad 8^2 = 64.
$$
Summing these gives $36 + 64 = 100$.
Thus the sum of the squares of the remaining five observations is:
$$
x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 560 - 100 = 460.
$$
Step 5: Compute the Variance of the Remaining 5 Observations
The variance of $x_1, x_2, x_3, x_4, x_5$ is given by:
$$
\text{Variance of 5 observations} = \frac{x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2}{5} \;-\;
\left( \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} \right)^2.
$$
Substituting the known sums:
$$
\frac{460}{5} - \left(\frac{42}{5}\right)^2.
$$
Simplify step by step:
$$
\frac{460}{5} = 92,
$$
and
$$
\left(\frac{42}{5}\right)^2 = \frac{1764}{25}.
$$
Converting 92 to a fraction with denominator 25:
$$
92 = \frac{92 \times 25}{25} = \frac{2300}{25}.
$$
Hence the variance becomes:
$$
\frac{2300}{25} - \frac{1764}{25} = \frac{536}{25}.
$$
Step 6: State the Final Answer
The variance of the remaining 5 observations is
$$
\frac{536}{25}.
$$
