Question

The GRE is a standardized test that students usually take before entering graduate school. According to a document, the scores on the verbal portion of the GRE have mean 150 points and standard deviation 8.75 points. Assuming that these scores are (approximately) normally distributed.

a. Obtain and interpret the quartiles.

b. Find and interpret the 99th percentile.

This has already been answered but one of the first steps includes .67 and I don't understand where .67 came from. My main question isn't so much find the answer, but go into more detail about where .67 comes from. Thank you!

Answer 1

Solution:

Given:  , the scores on the verbal portion of the GRE have mean 150 points and standard deviation 8.75 points.

The scores are (approximately) normally distributed.

Part a)  Obtain and interpret the quartiles.

Quartiles are the position values of the variable. There are three quartiles which divides whole data in 4 equal parts.

that is: each part constitute 25% of the data.

25% of the data is below first quartile Q1.

25%+25% = 50% of the data is below second quartile Q2.

25%+25% +25%= 75% of the data is below third quartile Q3.

Thus to obtain Q1,Q2 and Q3, we need to find z values corresponding to area shown above,

Thus find Q1 such that:

P(X < Q1) =25%

P(X < Q1) =0.25

Thus we need to find z value such that:

P( Z < z ) = 0.25

Look in z  table for Area = 0.2500 or its closest area and find corresponding z value.

Area 0.2514 is closest to 0.2500 and it corresponds to -0.6 and 0.07

that is  z = -0.67

Now use following formula to find Q1 value:

.

Now find Q2:

Thus find Q2 such that:

P(X < Q2) =50%

P(X < Q2) =0.50

Thus we need to find z value such that:

P( Z < z ) = 0.50

Look in z  table for Area = 0.5000 or its closest area and find corresponding z value.

Area 0.5000 corresponds to z = 0.0 and 0.00

Thus z = 0.00

Now use following formula to find Q2 value:

and find Q3 such that:

P(X < Q3) =75%

P(X < Q3) =0.75

Thus we need to find z value such that:

P( Z < z ) = 0.75

Look in z  table for Area = 0.7500 or its closest area and find corresponding z value.

Area 0.7486 is closest to 0.7500 and it corresponds to 0.6 and 0.07

that is  z = 0.67

Now use following formula to find Q3 value:

Thus we get:

25% of the scores on the verbal portion of the GRE are below first quartile 144.14

50% of the scores on the verbal portion of the GRE are below second quartile 150

75% of the scores on the verbal portion of the GRE are below third quartile 155.86