 # To be accepted into an honor society, a student must have a grade-point average (GPA) in the top 10% of the school. If the mean GPA at the school is 2.78 and the standard deviation is 0.33, which of the following GPAs would qualify? 3.20 , 3.21 , 3.25 , 3.30 , 3.35 ### Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836 ### Essentials Of Statistics

4th Edition
HEALEY + 1 other
Publisher: Cengage Learning,
ISBN: 9781305093836

#### Solutions

Chapter
Section
Chapter 5, Problem 5.12P
Textbook Problem
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## To be accepted into an honor society, a student must have a grade-point average (GPA) in the top 10% of the school. If the mean GPA at the school is 2.78 and the standard deviation is 0.33, which of the following GPAs would qualify? 3.20 , 3.21 , 3.25 , 3.30 , 3.35

Expert Solution
To determine

To find:

The qualified GPA.

### Explanation of Solution

Given:

The mean GPA is 2.78 and the standard deviation is 0.33.

Description:

The normal curve is symmetrical and its mean is equal to the median. The area below and above the mean is 50% or 0.05.

Formula used:

Let the data values be denoted by Xi.

The formula to calculate the Z score is given by,

Z=XiX¯s

Where, X¯ is the mean and s is the standard deviation of the distribution.

Calculation:

Given that the mean is 2.78 and standard deviation is 0.33.

For the GPA of 3.20,

Substitute 3.20 for Xi, 2.78 for X¯, and 0.33 for s in the above mentioned formula,

Z=3.202.780.33=0.420.33=1.27

The area between mean and the score 1.27 is 0.398.

Use the concept of symmetry, the area below the score 1.27 is,

0.3980+0.5=0.8980

The probability that the GPA is below 3.20 is 0.898.

For the GPA of 3.21,

Substitute 3.21 for Xi, 2.78 for X¯, and 0.33 for s in the above mentioned formula,

Z=3.212.780.33=0.430.33=1.30

The area between mean and the score 1.30 is 0.4032.

Use the concept of symmetry, the area below the score 1.30 is,

0.4032+0.5=0.9032

The probability that the GPA is below 3.21 is 0.9032.

For the GPA of 3.25,

Substitute 3.25 for Xi, 2.78 for X¯, and 0

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