## Question 516:

1

So we need to work backwards to find the z-score given the percentile. We know the formula for a z-score is (datpoint-mean)/SD. This gets us the z-score, but we have a percentile. First we need to convert the percentile to the z-score which we do by looking up the value in a normal table or using the percentile to z-score calculator (select 1-sided area), this gets us a z of -0.7386.

1. (x-70)/15 = -0.7386
2. x-70 = -11.079
3. x = 58.921

Finally, for the t-score, this could be one of two things. Typically the t-score refers to the t-distribution which is like the normal except takes into account sample size. Since we don't have sample size given, we cannot compute this t-score. The other approach used often in psychometrics (personality tests etc) is to convert the z-score to a positive number. This distribution has a mean of 50 and standard deviation of 10. The standard normal distribution, which is what the z-score is based on, involves converting the data to a distribution with a mean of 0 and standard deviation of 1.

The formula is t = 10(z)+50

We just need the z-score we found earlier. Filling in what we have we get t = 10(-0.7386) +50 = 42.614

So the z-score we found was -0.7386 , the raw score is 58.921 and the t-score is 42.614.