Math  /  Data & Statistics

QuestionGRE scores follow a normal distribution with a mean of of 1000 and a standard deivation of 200. Suppose we choose a student at random.
What is the probability that the student scores between 900 and 1300?1300 ?

Studdy Solution

STEP 1

1. GRE scores are normally distributed with a mean (μ\mu) of 1000 and a standard deviation (σ\sigma) of 200.
2. We need to find the probability that a score falls between 900 and 1300.

STEP 2

1. Convert the GRE scores to standard normal zz-scores.
2. Use the standard normal distribution to find the probability corresponding to these zz-scores.
3. Calculate the probability that the score is between the two zz-scores.

STEP 3

Calculate the zz-score for 900 using the formula:
z=Xμσ z = \frac{X - \mu}{\sigma}
For X=900X = 900:
z=9001000200=100200=0.5 z = \frac{900 - 1000}{200} = \frac{-100}{200} = -0.5

STEP 4

Calculate the zz-score for 1300 using the same formula:
For X=1300X = 1300:
z=13001000200=300200=1.5 z = \frac{1300 - 1000}{200} = \frac{300}{200} = 1.5

STEP 5

Use the standard normal distribution table (or a calculator) to find the probability for z=0.5z = -0.5 and z=1.5z = 1.5.
For z=0.5z = -0.5, the cumulative probability is approximately 0.3085.
For z=1.5z = 1.5, the cumulative probability is approximately 0.9332.

STEP 6

Calculate the probability that the score is between 900 and 1300 by subtracting the cumulative probability at z=0.5z = -0.5 from the cumulative probability at z=1.5z = 1.5:
P(900<X<1300)=P(z<1.5)P(z<0.5)=0.93320.3085=0.6247 P(900 < X < 1300) = P(z < 1.5) - P(z < -0.5) = 0.9332 - 0.3085 = 0.6247
The probability that a student scores between 900 and 1300 is:
0.6247 \boxed{0.6247}

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