Question

If the mean and variance of the numbers $2, 4, 6, 8, 10$ are $\mu$ and $\sigma^2$ respectively, then the value of $\mu + \sigma^2$ is:

• A. $12$
• B. $14$
• C. $16$
• D. $18$

Hint:

For equally spaced numbers symmetric about the center, the mean is always the middle term.

Answer 1

The Correct Option is B

Concept: The mean of a set of observations is the average of all values, while the variance measures the spread of the observations around the mean. For $n$ observations $x_1,x_2,\dots,x_n$: \[ \mu=\frac{\sum x_i}{n} \] and \[ \sigma^2=\frac{\sum (x_i-\mu)^2}{n} \]

Step 1: Finding the mean of the given observations. The observations are: \[ 2,\ 4,\ 6,\ 8,\ 10 \] Their sum is: \[ 2+4+6+8+10=30 \] Number of observations: \[ n=5 \] Therefore, \[ \mu=\frac{30}{5}=6 \]

Step 2: Calculating deviations from the mean. The deviations from the mean are: \[ 2-6=-4 \] \[ 4-6=-2 \] \[ 6-6=0 \] \[ 8-6=2 \] \[ 10-6=4 \]

Step 3: Squaring the deviations. \[ (-4)^2=16 \] \[ (-2)^2=4 \] \[ 0^2=0 \] \[ 2^2=4 \] \[ 4^2=16 \] Sum of squared deviations: \[ 16+4+0+4+16=40 \]

Step 4: Finding the variance. \[ \sigma^2=\frac{40}{5}=8 \]

Step 5: Finding the required value. \[ \mu+\sigma^2=6+8=14 \] \[ \boxed{14} \]