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Step-by-Step Solution
Step 1: Identify the incorrect and correct sums
The mean of the 20 observations was initially calculated as 10. This implies:
$$
\overline{x} = 10 \quad \Longrightarrow \quad \frac{\sum x_i}{20} = 10 \quad \Longrightarrow \quad \sum x_i = 200.
$$
It was later discovered that one observation was taken as 25 instead of 35. Hence, the correct sum of all observations should be:
$$
200 - 25 + 35 = 210.
$$
Step 2: Compute the correct mean
Using the corrected sum of observations, the correct mean $ \alpha $ is:
$$
\alpha = \frac{210}{20} = 10.5.
$$
Step 3: Use the given standard deviation to find incorrect sum of squares
The original (incorrect) standard deviation was given as 2.5, which means:
$$
\sigma = 2.5 \quad \Longrightarrow \quad \sigma^2 = 2.5^2 = 6.25.
$$
Recall the formula for variance:
$$
\sigma^2 = \frac{\sum x_i^2}{n} - (\overline{x})^2,
$$
where $n = 20$ and the initially used mean was $10$. Substituting these values:
$$
6.25 = \frac{\sum x_i^2}{20} - 10^2
\quad \Longrightarrow \quad
\frac{\sum x_i^2}{20} = 6.25 + 100 = 106.25
\quad \Longrightarrow \quad
\sum x_i^2 = 20 \times 106.25 = 2125.
$$
This $ \sum x_i^2 = 2125 $ is based on the incorrect data.
Step 4: Find the correct sum of squares
Since the incorrect data of 25 should have been 35, we must adjust the sum of squares accordingly:
$$
\sum x_i^2 \ (\text{correct})
= 2125 - 25^2 + 35^2
= 2125 - 625 + 1225
= 2725.
$$
Step 5: Compute the correct variance and standard deviation
Using the corrected sum of squares, the correct variance is:
$$
\sigma_{\text{correct}}^2
= \frac{2725}{20} - (10.5)^2
= 136.25 - 110.25
= 26.
$$
Hence,
$$
\sqrt{\beta} = \sigma_{\text{correct}} = \sqrt{26},
\quad \text{so} \quad
\beta = 26.
$$
Step 6: State the final answer
Therefore, the correct mean and variance are:
$$
\alpha = 10.5, \quad \beta = 26.
$$
This matches the given correct answer, $(10.5, 26)$.
