Please help answer with steps on how to do it.

Question

The grade point averages (GPAs) of all college students are normally distributed with mean 2.1 and standard deviation 0.5.

Part A

About  % of all students have the GPA at least 1.6.

(Rounding the percentage to 2 decimal places if possible)

Part B

About  % of all students have the GPA between 0.6 and 1.6.

(Rounding the percentage to 2 decimal places if possible)

Part C

About  % of all students have the GPA lower than 2.6.

(Rounding the percentage to 2 decimal places if possible)

Part D

About middle 60% of all students the GPA between  and  .

(Rounding the answer to 3 decimal places if possible)

Part E

About 40% of all students have the GPA higher than  .

(Rounding the answer to 3 decimal places if possible)

Part F

About 91% of all students have the GPA lower than  .

(Rounding the answer to 3 decimal places if possible)

Answer 1

Hi Christian, See below for some of the questions. With this, i think you can do the rest! Let me know if you have more questions.

A. 1.6 = 2.1-0.5 = μ - σ

P(X > μ - σ) =?

Convert to Z to solve the equivalent problem: ((μ - σ)-μ)/σ = -1

P(Z>-1) = 1-P(Z< -1) =1- 0.1587 = 0.841 ~ 0.84, which we got by reading the Z score table

To help our intuition, have in mind the empirical rule or the 68-95-99.7 rule (if it helps you, have its picture in front of you as you are solving this problem):

P(μ - σ < X < μ + σ) = 0.682

And with the normal distribution being symmetrical:

P( μ - σ < X < μ ) = 0.682/2 = 0.341

Thus, we can break down as: 

P(X> μ - σ) = P(X>μ ) + P ( μ - σ<X< μ ) = 0.5 + 0.341 = 0.841 ~ 0.84

B.

P (μ - 3σ < X < μ - σ) = ?

P (μ - 3σ< X < μ - σ) = 1 - P(X< μ - 3σ) - P(X> μ - σ) =  ?

Well, we know P(X> μ - σ) = 0.84 from part A above.

P(X<μ - 3σ) = P(Z<-3) = 0.0013

Thus:

P (μ - 3σ< X < μ - σ) = 1 - 0.0013-0.84 = 0.158~0.16

D. 

Now we will read the Z score tables from the inside out since we are interested to find the value that will give us a probability of 0.4.

P(X>V) = 0.4

This means that P(X<V) = 0.6. We read 0.26 as corresponding to 0.6 and thus P(Z< 0.26) = 0.6

(V- μ )/ σ = 0.26

Plugging in we get:

V -μ = 0.13, thus V = 2.23