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

The interval
`(300,700)` corresponds to the part of the distribution lying within 2 standard deviations of the mean (since 500-2*100=300 and 500+2*100=700). The empirical rule states that approximately 95% of the distribution is expected to fall in this range.

Answer 2

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

Explanation:

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

To find : What is the probability that a randomly selected student’s math score is between 300 and 700?

Solution :

The mean is
`\mu=500`

The standard deviation is
`\sigma=100`

Formula to find z-score is

`z=(x-\mu)/(\sigma)`

Now, we have to find the probability that a randomly selected student’s math score is between 300 and 700

Substitute x = 300 in the formula,

`z = (300-500)/(100)`

`z =-2`

Substitute x = 700 in the formula,

`z = (700-500)/(100)`

`z =2`

So, the probability between
`P(-2<z<2)`

`P(z<2)-P(z<-2)`

Using the z table substitute the values of z at -2 and 2.

P=0.9772-0.0228

P=0.9544

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