A state administered standardized reading exam is given to eighth grade students. The scores on this exam for all students statewide have a normal distribution with a mean of 507 and a standard deviation of 60. A local Junior High principal has decided to give an award to any student who scores in the top 10% of statewide scores.
How high should a student score be to win this award?
584
To find out how high a student should score in order to win the award, we need to calculate the score that corresponds to the top 10% of statewide scores.
First, let's find the z-score corresponding to the top 10% of scores. The z-score represents the number of standard deviations a particular score is from the mean.
Since the scores have a normal distribution, we can use the z-score formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value we want to find (the score we're looking for)
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
In this case, the mean (μ) is 507 and the standard deviation (σ) is 60.
To find the z-score corresponding to the top 10%, we need to find the z-score that leaves a total of 90% of the scores below it. We can look this up in a standard normal distribution table or use a calculator.
According to the standard normal distribution table, the z-score that leaves 90% of the scores below it is approximately 1.28.
Now we can use the z-score formula to find the corresponding score (x):
1.28 = (x - 507) / 60
Simplifying the equation:
1.28 * 60 = x - 507
76.8 = x - 507
x = 507 + 76.8
x = 583.8
So, a student would need to score approximately 583.8 or higher to win this award.