Calculating Correct Mean After Input Errors

The problem involves finding the correct mean of a set of numbers when some values used in the initial calculation were incorrect.

Initial Calculation Summary

• Number of observations ($n$): 80
• Incorrect Mean ($\bar{x}_{incorrect}$): 49

Step 1: Calculate the Initial Sum

The sum of the numbers initially calculated is the product of the incorrect mean and the number of observations.

Initial Sum = Incorrect Mean $\times$ Number of observations

Sum$_{initial}$ = $\sum_{i=1}^{80} x_{incorrect, i} = \bar{x}_{incorrect} \times n = 49 \times 80 = 3920$

Step 2: Identify Sum of Incorrect and Correct Values

The values that were incorrectly used and the correct values are given.

• Incorrect numbers: 69 and 74
• Sum of incorrect numbers = $69 + 74 = 143$
• Correct numbers: 92 and 83
• Sum of correct numbers = $92 + 83 = 175$

Step 3: Calculate the Correct Sum

To find the correct sum, subtract the sum of the incorrect numbers from the initial sum and add the sum of the correct numbers.

Correct Sum = Initial Sum - Sum of incorrect numbers + Sum of correct numbers

Sum$_{correct}$ = $3920 - 143 + 175$

Sum$_{correct}$ = $3920 + (175 - 143)$

Sum$_{correct}$ = $3920 + 32 = 3952$

Step 4: Calculate the Correct Mean

The correct mean is calculated by dividing the correct sum by the total number of observations.

Correct Mean = Correct Sum / Number of observations

$\bar{x}_{correct} = \frac{\text{Sum}_{correct}}{n} = \frac{3952}{80}$

$\bar{x}_{correct} = 49.4$

Therefore, the correct mean is 49.4.