**Correct Answer: E**

**Step 1: Analyze Statement 1 alone**

Statement 1 tells us $x^2 = 16$, which means $x$ can be $4$ or $-4$. However, we have no information regarding the value or nature of $y$. If $y=1$, $x^y$ is an integer ($4$ or $-4$). If $y=0.5$, $x^y$ would be $\sqrt{4}=2$ (integer) or $\sqrt{-4}$ (not a real number, though the prompt states $y$ is a real number). If $y=-1$, $x^y$ could be $1/4$ (not an integer). Statement 1 is **not sufficient**.

**Step 2: Analyze Statement 2 alone**

Statement 2 tells us $y^2 = 4$, which means $y$ can be $2$ or $-2$. We have no information regarding $x$. If $x=2$, then $2^2=4$ (integer) or $2^{-2}=1/4$ (not an integer). Statement 2 is **not sufficient**.

**Step 3: Analyze Statements 1 and 2 together**

Combining the statements, we have $x \in \{4, -4\}$ and $y \in \{2, -2\}$.

Let's test the possible outcomes for $x^y$:

1. $4^2 = 16$ (Integer)

2. $(-4)^2 = 16$ (Integer)

3. $4^{-2} = 1/16$ (Not an integer)

4. $(-4)^{-2} = 1/16$ (Not an integer)

Because we still get both "Yes" and "No" answers depending on whether $y$ is positive or negative, the statements together are **not sufficient**.

**Logical Trap:**

The most common error here is "Sign Oversight." Students often focus on the base ($x$) being a perfect square or an integer and assume that since both $x$ and $y$ have "clean" integer square roots, the result must be an integer. However, in exponentiation, the sign of the exponent ($y$) is the critical factor. A negative exponent transforms an integer base into a fraction, which—unless the base is $1$ or $-1$—will never be an integer.

**Conclusion:**

Since $y$ can be $-2$, the expression $x^y$ can result in a non-integer fraction. Without knowing the sign of $y$, we cannot definitively answer the question. Therefore, even together, the statements are insufficient.