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Step-by-step Solution
Step 1: Understand the given information
We have five observations whose mean is 9 and standard deviation (s.d.) is 0. Because the standard deviation is 0, it indicates that all five observations are exactly the same.
Step 2: Determine the original observations
Since each of the five observations is the same and their mean is 9, each observation must be 9. Mathematically, if x1, x2, x3, x4, x5 are the five observations, then:
$ \sum x_{i} = 9 \times 5 = 45 $
and each xi = 9.
Step 3: Incorporate the change in one observation
One observation is replaced such that the mean of the new set of five observations becomes 10. Let the new observation be x5. Then:
$ \frac{(9 + 9 + 9 + 9) + x_{5}}{5} = 10
$
Note that we replaced only one observation, so the other four remain at 9.
Step 4: Solve for the new observation
$ \frac{36 + x_{5}}{5} = 10 \\
36 + x_{5} = 50 \\
x_{5} = 14
$
Step 5: Compute the new standard deviation
The new set of observations is (9, 9, 9, 9, 14). The mean of this new set is 10. The standard deviation is given by:
$ \sigma_{\text{new}} = \sqrt{\frac{\sum (x_{i} - \overline{x}_{\text{new}})^2}{5}}
$
Let xi be each of the new observations and $ \overline{x}_{\text{new}} $ = 10. Then:
$ \sigma_{\text{new}}
= \sqrt{\frac{(9 - 10)^2 + (9 - 10)^2 + (9 - 10)^2 + (9 - 10)^2 + (14 - 10)^2}{5}} \\
= \sqrt{\frac{4 \times ( -1 )^2 + (4)^2}{5}} \\
= \sqrt{\frac{4 \times 1 + 16}{5}} \\
= \sqrt{\frac{4 + 16}{5}} \\
= \sqrt{\frac{20}{5}} \\
= \sqrt{4} \\
= 2
$
Final Answer
The new standard deviation of the five observations is 2.
