Let $X = \dfrac{1}{25} \sum\limits_{i=1}^{25} X_i$ and $Y =\dfrac{5}{2}X - \dfrac{2}{5}$. What is $P(|Y| > 1)$? 
Suppose that $X_1,X_2,\ldots,X_{25}$ are independent random variables from $\mathcal{N}(1, 4)$. Let $X = \dfrac{1}{25} \sum\limits_{i=1}^{25} X_i$ and $Y =\dfrac{5}{2}X - \dfrac{2}{5}$. What is the probability that $|Y| > 1$? 

We know that $Y$ is a standard normal random variable.
I keep trying to transform it into normal distribution, but I keep getting the wrong answer. The answer is $0.6826$. How do I solve this?

$Y$ is not a standard normal random variable!
If $X_i \sim N(1,4)$, then $X=\frac{1}{25}\sum_{i=1}^{25} X_i \sim N(1,\frac{4}{25}).$ So $Y$ is normal with mean 
$$EY=\frac{5}{2} -\frac{2}{5}=\frac{21}{10},$$ 
and  variance 
$$\textrm{var}(Y)=\left(\frac{5}{2}\right)^2 \cdot  \frac{4}{25} =1.$$
Can you finish it?  
