Question

A test was given in which the mean grade was 70% and the standard deviation was 12. a) If a student has a standard score (z) of -2.74, what is the student grade on the test?

b) If the student standard score (z) is 1.34, what is the student score on the test?
c) what is the student percentile?
d) What percent of people score higher and lower than the student grade?

Answer 1

`Z`-score `-2.74` means a score of `37.12`,
`Z`-score `1.34` means a score of `86.08`,
Percentile for `z=-2.74` is `0.307` and for `z=1.34` it is `90.988`
Percentage of people above `z=-2.74` is `99.693` and those above `z=1.34` is `9.012`

Explanation:

`Z`-score is given by `z=(x-mu)/sigma` i.e. `x=zxxsigma+mu`, where `mu` is mean and `sigma` is standard deviation,

Here mean `mu=70` and standard deviation `sigma=12`

Hence, a `Z`-score `-2.74` means a score of `(-2.74)xx12+70`

= `-32.88+70=37.12`

And a `Z`-score `1.34` means a score of `1.34xx12+70`

= `16.08+70=86.08`

From the Standard Normal tables, area to the left of `z=-2.74` is `0.00307`, Hence percentile for `z=-2.74` is `0.307` and for `z=1.34` it is `100-9.012=90.988`.

Hence, percentage of people above `z=-2.74` is `100-0.307=99.693` and percentage of people above `z=1.34` is `9.012`