EXAMPLE 40Finding a normal data value for a given area
SAT Scores and AP Exam Scores
Suppose the students in the top 1% of Math SAT scores won a fellowship to an Ivy League university. What is the score that students will have to obtain to win this fellowship?
Solution
Notice that we are not asked to find a probability (or proportion or area). Instead, we are given a percentage (1%) and asked to find the value of (the Math SAT score) that is associated with this 1%.
Step 1 Determine , , and , and draw the normal curve for .
We already know that , , and . The value of in which we are interested refers to high scores, so that will be at the far right of the distribution of . Only 1% of scores will be greater than this score, so the area to the right of is 0.01, as shown in Figure 52.
Figure 6.54: FIGURE 52 is the cutoff value (or critical value) of , at which students will win a fellowship to an Ivy League university.373
Step 2 Find the -value corresponding to the desired area.
The area to the right of equals 0.01, so that the area to the left of equals . Looking up 0.99 on the inside of the table gives us .
Step 3 Transform using the formula .
We calculate
The cutoff value for the top 1% of Math SAT scores for winning a fellowship to an Ivy League university is 788.94. It won't be easy getting that fellowship.
NOW YOU CAN DO
Exercises 15–22.