The mean of 150 observations was found to be 45. If at the time of calculation, two items were wrongly taken as 42 and 28 instead of 35 and 25, then find the correct mean.

Options

A. 46
B. 44.9
C. 45.9
D. 43.5

Answer

To find the correct mean after correcting the two wrongly taken observations, we can follow these steps:

### Step 1: Calculate the sum of the original observations
The mean of 150 observations is given as 45. We can calculate the sum of these observations using the formula for mean:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Thus, the sum of the observations (denoted as \( \Sigma x \)) can be calculated as:

\[ \Sigma x = \text{Mean} \times \text{Number of observations} = 45 \times 150 = 6750 \]

### Step 2: Identify the incorrect and correct observations
The two observations that were incorrectly recorded are 42 and 28. The correct observations should have been 35 and 25.

### Step 3: Calculate the difference caused by the incorrect observations
Now, we need to adjust the sum of the observations by removing the incorrect values and adding the correct values:

1. **Remove the incorrect observations:**
\[ \text{Sum after removing incorrect observations} = 6750 - (42 + 28) = 6750 - 70 = 6680 \]

2. **Add the correct observations:**
\[ \text{Sum after adding correct observations} = 6680 + (35 + 25) = 6680 + 60 = 6740 \]

### Step 4: Calculate the correct mean
Now that we have the corrected sum of observations, we can find the new mean:

\[ \text{Correct Mean} = \frac{\text{Correct Sum}}{\text{Number of observations}} = \frac{6740}{150} \]

Calculating this gives:

\[ \text{Correct Mean} = 44.9333 \ldots \approx 44.93 \]

### Step 5: Round the mean to one decimal place
Rounding to one decimal place, we get:

\[ \text{Correct Mean} \approx 44.9 \]

Thus, the correct mean is **44.9**.

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