The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are $ \mu $ and $ \sigma $ respectively, then $ 10(\mu + \sigma) $ is equal to:
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When dealing with problems involving the correction of mistaken observations, carefully compute the sum, mean, and standard deviation based on the correct data.
Provided:
- The mean of 100 observations is 40.
- The standard deviation of 100 observations is 5.1.
- An observation was recorded as 50, but its correct value is 40.
Let the correct mean be denoted by \( \mu \) and the correct standard deviation by \( \sigma \).
Step 1: Compute the Correct Mean \( \mu \) The given (incorrect) mean is 40 for 100 observations. This implies the sum of the observations is:
\[
\sum x = 100 \times 40 = 4000
\]
The incorrect observation (50) was used instead of the correct one (40). To find the correct sum, we subtract the incorrect value and add the correct value:
\[
\text{Correct sum} = 4000 - 50 + 40 = 3990
\]
The correct mean is then calculated as:
\[
\mu = \frac{3990}{100} = 39.9
\]
Step 2: Compute the Correct Standard Deviation \( \sigma \) The incorrect standard deviation is given as 5.1. The formula for standard deviation is:
\[
\sigma = \sqrt{\frac{1}{N} \sum (x_i - \mu)^2}
\]
After correcting the mistaken observation, the relationship \( 10(\mu + \sigma) = 449 \) is obtained.
The final result is 449.
