1. Given data for the 50 observations (with one wrong observation included):

• Number of observations: $n=50$
• Mean: $\bar x=15$
• Standard deviation: $\sigma=2$

So the total sum is $\sum x = n\bar x = 50\cdot 15 = 750.$

Also, $\sigma^2 = 4.$ Using $\sigma^2 = \frac{\sum x^2}{n} - \bar x^2,$ we get $4 = \frac{\sum x^2}{50} - 15^2.$ Thus, $4 = \frac{\sum x^2}{50} - 225$ $\frac{\sum x^2}{50} = 229$ $\sum x^2 = 50\cdot 229 = 11450.$

2. Use the condition about correct and incorrect observation:

Let the incorrect observation be $w$ and the correct observation be $c$. Given: $w+c=70.$

Also, the correct mean is $16$, so the correct total sum should be $50\cdot 16 = 800.$

Now, $\text{correct sum} = \text{wrong sum} - w + c$ $800 = 750 - w + c$ $c-w=50.$

Now solve the system: $c+w=70,\qquad c-w=50.$ Adding, $2c=120 \Rightarrow c=60.$ Then, $w=10.$

3. Find the correct sum of squares:

The wrong sum of squares is $11450.$ Replace $w=10$ by $c=60$: $\sum x^2_{\text{correct}} = 11450 - 10^2 + 60^2$ $=11450 - 100 + 3600$ $=14950.$

4. Compute the correct variance:

For the corrected data,

• Mean $=16$
• $n=50$
• $\sum x^2=14950$

Hence variance is $\sigma^2 = \frac{14950}{50} - 16^2$ $=299 - 256$ $=43.$

5. Match with options:

The correct variance is $\boxed{43}.$ So the correct option is C.