Question

5.4 GPA is approximately normally distributed for all students that have been accepted to UCLA. The average gpa is 4.0 with standard deviation of .2. 50 students were sampled and asked, "what is your gpa?":

a) Calculate the probability the mean gpa is 3.95 or lower

b) Calculate the standard deviation for the sampling distribution.

b) Between what 2 gpa's are considered normal for being accepted at UCLA

c) What gpa represents the top 10% of all students

d) What percent had a gpa above 4.1

e) What % of admitted students had a gpa lower than 3.5

Answer 1

Solution:-

Mean = 4.0, S.D = 0.20, n = 50

a) The probability the mean gpa is 3.95 or lower is 0.0385.

x = 3.95

By applying normal distribution:-

z = - 1.7678

Use the z-score table or p-value calculator.

P(z < - 1.7678) = 0.0385

b) The standard deviation for the sampling distribution is 0.0283

s = 0.0283

b) 2 gpa's are considered normal for being accepted at UCLA are 3.86 and 4.14.

The gpa's between 2 standard deviations are considered normal.

z₁ = - 2.0

z₂ = 2.0

By applying normal distribution:-

x₁ = 3.86

x₂ = 4.14

c) The gpa represents the top 10% of all students is 4.0363.

p-value for the top 10% = 1 - 0.10 = 0.90

z-score for the p-value = 1.282

By applying normal distribution:-

x = 4.0363

d) 0.02% had a gpa above 4.1.

x = 4.10

By applying normal distribution:-

z = 3.54

Use the z-score table or p-value calculator.

P(z < 3.54) = 0.0002

P(z < 3.54) = 0.02%

e) 0 % of admitted students had a gpa lower than 3.5.

x = 3.50

By applying normal distribution:-

z = - 17.68

Use the z-score table or p-value calculator.

P(z < - 17.68) = 0.000

P(z < - 17.68) = 0.0%