Probability of Normal Distribution

[needhelp83]

Test scores on a standardized test have mean 55 and variance 64. What is the probability that a random sample of 256 scores will have a sample mean score between 53 and 57

I attempted the problem and came up with this:
u=55 and sd = 8 (Square root of 64)
$$\mu=55 \sigma=\sqrt{64}=8$$
$$P(53< \bar{X} < 57) = P(\frac{53-55}{\frac{8}{\sqrt{256}}}<Z<\frac{57-55}{\frac{8}{\sqrt{256}}})=P(-4<Z<4)$$

What am I doing wrong?

 

[vela]

Your work looks fine to me. Why do you think there's something wrong?

 

[needhelp83]

I felt like it was off because 4 is off the charts, so I figured I managed to perform a miscalculation somewhere

 

[vela]

Yeah, I know the feeling. These statistics and probability problems can really test your intuition quite a bit.

 

[needhelp83]

$$\Phi(4)-\Phi(-4) = 1 - 0 =1$$

So is this correct?

 

[vela]

Yes, to the accuracy of four decimal places. The large sample size really narrows down the uncertainty in the mean.

 

[needhelp83]

Ok, thanks for the explanation. That makes sense