Probability of Normal Distribution

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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)
[tex]\mu=55 \sigma=\sqrt{64}=8[/tex]
[tex]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)[/tex]

What am I doing wrong?
 

Answers and Replies

  • #2
vela
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Your work looks fine to me. Why do you think there's something wrong?
 
  • #3
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I felt like it was off because 4 is off the charts, so I figured I managed to perform a miscalculation somewhere
 
  • #4
vela
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Yeah, I know the feeling. These statistics and probability problems can really test your intuition quite a bit.
 
  • #5
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[tex]\Phi(4)-\Phi(-4) = 1 - 0 =1[/tex]

So is this correct?
 
  • #6
vela
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Yes, to the accuracy of four decimal places. The large sample size really narrows down the uncertainty in the mean.
 
  • #7
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Ok, thanks for the explanation. That makes sense
 

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