Question

4. Dean Pelton wants to perform calculations to impress the accreditation consultants, but upon asking for information about GPAs at Greendale Community College, Chang only tells Pelton that the GPAs are distributed with a probability density function f(x) = D(2 + e −x ), 2 ≤ x ≤ 4 where D was some unknown "Duncan" constant. How many student records have to be retrieved so that the probability that the average GPA is less than 2.3 is less than 4 percent?

Answer 1

the minimum records to be retrieved by using Chebysher - one sided inequality is 17.

Explanation:

Let assume that n should represent the number of the students

SO,
`\bar x` can now be the sample mean of number of students in GPA's

To obtain n such that
`P( \bar x \leq 2.3 ) \leq .04`

⇒
`P( \bar x \geq 2.3 ) \geq .96`

However ;

`E(x) = \int\limits^4_2 Dx (2+e^(-x) ) 4x = D \\ \\ = D(e^(-x) (e^xx^2 - x-1 ) ) ^D_2 = 12.314 D`

`E(x^2) = D\int\limits^4_2 (2+e^(-x))dx \\ \\ = (D)/(3)[e^(-4) (2e^x x^3 -3x^2 -6x -6)]^4__2}}= 38.21 \ D`

Similarly;

`D\int\limits^4_2(2+ e^(-x)) dx = 1`

⇒
`D*(2x-e^(-x) ) |^4_2 = 1`

⇒
`D*4.117 = 1`

⇒
`D= (1)/(4.117)`

`\mu = E(x) = 2.991013 ; \\ \\ E(x^2) = 9.28103`

∴
`Var (x) = E(x^2) - E^2(x) \\ \\ = .3348711`

Now;
`P(\bar \geq 2.3) = P( \bar x - 2.991013 \geq 2.3 - 2.991013) \\ \\ = P( \omega \geq .691013) \ \ \ \ \ \ \ \ \ \ (x = E(\bar x ) - \mu)`

Using Chebysher one sided inequality ; we have:

`P(\omega \geq -.691013) \geq ((.691013)^2)/(Var ( \omega) +(.691013)^2)`

So;
`(\omega = \bar x - \mu)`

⇒
`E(\omega ) = 0 \\ \\ Var (\omega ) = (Var (x_i))/(n)`

∴
`P(\omega \geq .691013) \geq ((.691013)^2)/((.3348711)/(n)+(691013)^2)`

To determine n; such that ;

`((.691013)^2)/((.3348711)/(n)+(691013)^2) \geq 0.96 \\ \\ \\ (.691013)^2(1-.96) \geq (-3348711*.96)/(n)`

⇒
`n \geq (.3348711*.96)/(.04*(.691013)^2)`

`n \geq 16.83125`

Thus; we can conclude that; the minimum records to be retrieved by using Chebysher - one sided inequality is 17.