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Step-by-Step Solution
Step 1: Write down the known information
• We have five observations in total.
• Their mean is 5.
• Their variance is 9.20.
• Three of these observations are 1, 3, and 8.
• We need to find the ratio of the remaining two observations, say $x_1$ and $x_2$.
Step 2: Express the condition from the mean
The mean of five observations is given by
$ \text{Mean} = \frac{x_1 + x_2 + 1 + 3 + 8}{5} = 5. $
Since the mean is 5, multiplying both sides by 5 gives
$ x_1 + x_2 + 1 + 3 + 8 = 25. $
Thus,
$ x_1 + x_2 = 25 - (1 + 3 + 8) = 25 - 12 = 13. \quad (1) $
Step 3: Express the condition from the variance
The variance of the five observations is 9.20. Recall the formula for variance ($\sigma^2$) when dealing with a set of values $x_1, x_2, \dots, x_n$ is:
$ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2. $
We know $\sum x_i = 25$ and the variance is 9.20. Let us write this explicitly:
$ \sigma^2 = 9.20 = \frac{\sum x_i^2}{5} - 5^2. $
Since $5^2 = 25$, we have:
$ 9.20 = \frac{\sum x_i^2}{5} - 25. $
Rearranging to find $\sum x_i^2$:
$ \frac{\sum x_i^2}{5} = 9.20 + 25 = 34.20, $
$ \sum x_i^2 = 34.20 \times 5 = 171. $
Hence, the sum of the squares of all five observations is 171. Because we already know three of the observations (1, 3, and 8), we calculate:
$1^2 + 3^2 + 8^2 = 1 + 9 + 64 = 74. $
Therefore,
$ x_1^2 + x_2^2 = 171 - 74 = 97. \quad (2) $
Step 4: Use identities to find the product $x_1 \times x_2$
From equation (1), $x_1 + x_2 = 13$, we square both sides:
$ (x_1 + x_2)^2 = 13^2 = 169. $
We also know that:
$ (x_1 + x_2)^2 = x_1^2 + 2 x_1 x_2 + x_2^2. $
Using $x_1^2 + x_2^2 = 97$ from (2), we substitute into the above:
$ 169 = (x_1^2 + x_2^2) + 2x_1 x_2 = 97 + 2 x_1 x_2. $
Rearranging gives:
$ 169 - 97 = 2 x_1 x_2 \quad \Rightarrow \quad 72 = 2 x_1 x_2 \quad \Rightarrow \quad x_1 x_2 = 36. $
Step 5: Determine the ratio of $x_1$ to $x_2$
We have two equations involving $x_1$ and $x_2$:
1) $x_1 + x_2 = 13$
2) $x_1 x_2 = 36$
These can be thought of as the system describing two numbers whose sum is 13 and product is 36. One such valid pair is (4, 9), because $4 + 9 = 13$ and $4 \times 9 = 36.$
Hence, the ratio of $x_1$ to $x_2$ is
$ x_1 : x_2 = 4 : 9. $
