How do I solve the following: In a normal distribution with a mean of 180 and a standard deviation of 30, what raw score falls at the 60th percentile (or P.60)?

We look on the Z chart for P=0.60. According to the Z chart, it gives a Z value of 0.25

So it's 0.25 standard deviations above the mean:

(0.25)(30) = 7.5

Therefore, the raw score is 180 + 7.5 = 187.5

When I find the Z value of 0.60 on the chart, it says 0.2257 next to it (area between mean and z). Am I misunderstanding how to read the chart?

yes, you are reading it wrong. the z value only has 2 significant numbers. In this case it is 0.25

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