Question

Grade:ABCDF

Probability:0.10.30.40.10.1

To calculate student grade point averages, grades are expressed in a numerical scale with A = 4, B = 3, and so on down to F = 0.

Find the expected value. This is the average grade in this course.

Explain how to simulate choosing students at random and recording their grades. Simulate 50 students and find the mean of their 50 grades. Compare this estimate of the expected value with the exact expected value from part (a). (The law of large numbers says that the estimate will be very accurate if we simulate a very large number of students.)

Answer 1

Calculation for expected value-

Suppose, random variable X denotes numerical value corresponding to grade obtained.

So, we have probability mass function as follows.

Grade obtained	A	B	C	D	F	Total
Numerical scale value,	4	3	2	1	0	-
Probability,	0.10	0.30	0.40	0.10	0.10	1

So, expected value

  

= 0*0.10 + 1*0.10 +2*0.40 + 3*0.30 +4*0.10 = 2.2

Simulation-

We can perform simulation either by hand using a random number table or by using any statistical software.

Let us simulate by hand using random number table.

X	P(X=x)	Cumulative probability	Random number interval
0	0.10	0.10	00-09
1	0.10	0.20	10-19
2	0.40	0.60	20-59
3	0.30	0.90	60-89
4	0.10	1	90-99

Note- As we are going to take last two digits we have taken interval 00-99 (instead of 01-100).

We now take last two digits of each number of the following random number table to use. One may use any other random number table or even this one considering any other way of choosing (e.g. first two digits of each random number).

Simulated numerical scale values are as follows.

Serial number	1	2	3	4	5	6	7	8	9	10
Random number	64	18	61	74	20	15	91	31	78	54
Numerical scale value	3	1	3	3	2	1	4	2	3	2

Serial number	11	12	13	14	15	16	17	18	19	20
Random number	38	82	58	49	82	92	36	66	83	81
Numerical scale value	2	3	2	2	3	4	2	3	3	3

Serial number	21	22	23	24	25	26	27	28	29	30
Random number	44	73	32	14	84	63	72	63	04	83
Numerical scale value	2	3	2	1	3	3	3	3	0	3

Serial number	31	32	33	34	35	36	37	38	39	40
Random number	08	37	01	17	42	86	75	05	41	49
Numerical scale value	0	2	0	1	2	3	3	0	2	2

Serial number	41	42	43	44	45	46	47	48	49	50
Random number	56	60	16	71	78	25	17	15	07	31
Numerical scale value	2	3	1	3	3	2	1	1	0	2

So, calculating number of occurrence of numerical scale values we get,

Numerical scale value	0	1	2	3	4	Total
Frequency	5	7	16	20	2	50

So, simulated mean = (0*5 + 1*7 +2*16 + 3*20 + 4*2) / 50 = 107 / 50 = 2.14

Conclusion-

We see expected mean (2.2) and simulated mean (2.14) are very close.

Note: If we simulate using another random number table or choosing number from this table with different procedure if choosing or using any statistical software, we will get different simulated mean values each time (even if we use same statistical software). But all these simulated values will be close to expected value.