Question

Part 1:

Suppose the mean score on the first exam is a 60 out of 100. I would like to curve the exam to force the mean score to be an 80 out of 100. If I added 20 to everyone’s score, would the SD of the exam scores change? How about if I multiplied 4/3 to everyone’s score? Explain.

Part 2: Continuing from Part 1, fill out the formulas put below that will always be true. Let a = constant (- or +) and X = random variable.

EV (a + X) =

SD (a + X) =

SD (aX) =

EV (aX) =

Variance (a + X) =

Variance (a + X) =

Answer 1

solution:

Part-1:

Let x1,x2,x3,......,Xn be the n observations

we know that     

   Mean = =

Standard Deviation =

Given that Before adding 20 to each observation

Mean =   = = 60

Standard Deviation =

After adding '20' to each observation

Mean1 = 1 =

=  

= + 20

1 = + 20

Standard Deviation =

=

=

SD remains same after adding 20 to each observation.

Now,

After  multiplying 4/3 to each observation

Mean =   = = 4/3 * = 4/3 *

Standard Deviation =

=

=

= 4/3 *

Standard Deviation increases by 4/3 times after multiplying

Part -2:

Let a be some constant and X be a random variable

Results from part-1:

1) If we add some K to each observation the mean increses by K and SD remains same

2)If we multiply each observation by K(constant) then mean and SD increases by K times

1) E(a+X) = E(a)+E(X) = a+E(X)

E(a+X) = a + E(X)

2) SD(a+X) = SD(X)

3) E(aX) = a*E(X)

4) Var(a+X) = Var(a) + Var(X) = Var(X)

5) Var(a+X) = Var(X)

Note: Var (a*X) = a^2 * Var(X)