Question

The SAT mathematics scores in the state of Florida are approximately normally distributed with a mean of 500 and a standard deviation of 100.

Using the empirical rule, what is the probability that a randomly selected student’s math score is between 300 and 700? Express your answer as a decimal.

Answer 1

0.9544 is the probability that a randomly selected student’s math score is between 300 and 700 .

Explanation:

Step 1: Sketch the curve.

The probability that 300<X<700 is equal to the blue area under the curve.

Step 2:

Since μ=500 and σ=100 we have:

P ( 300<X<700 )=P ( 300−500< X−μ<700−500 )

⇒P ( 300−500/100 < X−μ/σ < 700−500/ 100)

Since Z = x − μ /σ , 300−500 /100 = −2 and 700−500/100 = 2 we have:

P ( 300<X<700 )=P ( −2<Z<2 )

Step 3: Use the standard normal table to conclude that:

P ( −2<Z<2 )=0.9544

Therefore,0.9544 is the probability that a randomly selected student’s math score is between 300 and 700 .