# Stat 185¶

## The problem¶

The mean score for the Quantitative Reasoning section of the GRE is 153 with a standard deviation of 7.67.

1. If Adelaide scores a 165, then what proportion of students scored below Adelaide.

2. If Sophia scored a 150, then what proportion of students scored above Sophia.

3. What proportion of students scored between Adelaide and Sophia?

### Part 1¶

We first compute Adelaid's Z-score and find

(165-153)/7.67
1.5645371577574967

We then look 1.56 up in the $-\infty$ based version of our normal table and find

$$P(X<1.56) = 0.9406.$$

### Part 2¶

We now compute Sophia's Z-score and find

(150-153)/7.67
-0.39113428943937417

We then look -0.39 up in the $-\infty$ based version of our normal table and find

$$P(X<-0.39) = 0.3483.$$

The answer is thus $$1-0.3483 = 0.6517.$$

### Part 3¶

Now that we know for Adelaide that $$P(X<1.69) = 0.9406$$ and for Sophia that $$P(X<-0.39) = 0.3483,$$ the proportion of folks between must be $$0.9406 - 0.3483 = 0.5923.$$