If the mean of the observations : x+1,x+2,x+3,x+4,x+10 is 16, the mean of the first three observations isA) 20B) 17C) 14D) 9

1. Break Down the Problem

We have the observations: $x + 1$, $x + 2$, $x + 3$, $x + 4$, and $x + 10$. We know that their mean is 16. We need to find the mean of the first three observations: $x + 1$, $x + 2$, and $x + 3$.

2. Relevant Concepts

The mean of a set of numbers is given by the formula: $\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}$ For the five observations: $\text{Mean} = \frac{(x+1) + (x+2) + (x+3) + (x+4) + (x+10)}{5} = 16$

3. Analysis and Detail

Now, we calculate the sum of the observations: $(x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 10) = 5x + 20$ Setting this equal to the mean: $\frac{5x + 20}{5} = 16$ Multiplying both sides by 5: $5x + 20 = 80$ Now, we solve for $x$: $5x = 80 - 20$ $5x = 60$ $x = 12$

Next, we find the mean of the first three observations: $\text{Mean of } (x + 1), (x + 2), (x + 3) = \frac{(x + 1) + (x + 2) + (x + 3)}{3}$ Calculating the sum: $(x + 1) + (x + 2) + (x + 3) = 3x + 6$ Thus, the mean becomes: $\text{Mean} = \frac{3x + 6}{3} = x + 2$ Substituting $x = 12$: $\text{Mean} = 12 + 2 = 14$

4. Verify and Summarize

The mean of the first three observations is $14$.

Final Answer

The mean of the first three observations is $14$ (Option C).