mean score of 80 and a standard deviation of 5. Percentage of students that scored between 80 and 90?

Question

The scores on a certain math test were normally distributed with a mean score of 80 and a standard deviation of 5.  What percent of the students scored between 80 and 90?

Answer 1

If the mean (μ) is 80, and the standard deviation (σ) is 5, then all scores between 80 and 90 would fall between 0 and 2 standard deviations above the mean.

 

Using the equation for Z score (Z = (X-μ)/σ) for each X value (80 and 90) then the Z scores are 0 and 2, respectively.  

 

Using a normal distribution table, it can be found that P(80 < z) = .5 (this is the probability that a random score would be greater than 80.  It makes sense that it is .5 or 50% because 80 is the mean.)

And the P(90 > z) = .97725.  (this is the probability that a random score would be less than 90.)

 

So the final answers would be (90 > z)-P(80<z) = .47725 or 47.725%