Question

Simulate the effect of the Price change if it will follow the following pattern for Type A. (build the 95% confidence interval) Type A (Price (million Dollar) =1.25 (20% probability); Price (million Dollar) =2.25 (40 % probability); Price (million Dollar) =3 (25 % probability); Price (million Dollar) =3.5 (15 % probability))

Answer 1

Solution

NOTE: Question does not specify the number of simulation runs. As an arbitrary measure, 50 runs are made just to demonstrate the process and interpretation of results for further inferences………...............................................…… (1)

Back-up Theory

100(1 - α) % Confidence Interval for μ, when σ is not known is: Xbar ± (t_{n- 1, α /2})s/√n …………………………….…… (2)where

Xbar = sample mean, t_{n – 1, α /2} = upper (α /2)% point of

t-distribution with (n - 1) degrees of freedom, s = sample standard deviation and n = sample size.

Now to work out the solution,

Preparatory Work

Assignment of Random Numbers

Price ($106)	Probability	Cumulative Probability	Assigned Random Numbers
1.25	0.2	0.2	01 - 20
2.25	0.4	0.6	21 - 60
3	0.25	0.85	61 – 85
3.5	0.15	1.00	86 - 00

Random numbers used

Lines 10580–10594, columns 21–40, from

RAND Corporation - RAND's A Million Random Digits

73735 45963 78134 63873 02965 58303 90708 20025 98859 23851

27965 62394 33665 63570 64775 78428 81665 26440 20422 05720

Simulation Process

Run Number	Random Number	Price
1	73	3.00
2	73	3.00
3	54	2.25
4	59	2.25
5	63	3.00
6	78	3.00
7	13	1.25
8	46	2.25
9	38	2.25
10	73	3.00
11	02	1.25
12	96	3.5
13	55	2.25
14	83	3.00
15	03	1.25
16	90	3.5
17	70	3.00
18	82	3.00
19	00	3.5
20	25	2.25
21	98	3.5
22	85	3.00
23	92	3.5
24	38	2.25
25	51	2.25
26	27	2.25
27	96	3.5
28	56	2.25
29	23	2.25
30	94	3.5
31	33	2.25
32	66	3.00
33	56	2.25
34	35	2.25
35	70	3.00
36	64	3.00
37	77	3.00
38	57	2.25
39	84	3.00
40	28	2.25
41	81	3.00
42	66	3.00
43	52	2.25
44	64	3.00
45	40	2.25
46	20	1.25
47	42	2.25
48	20	1.25
49	57	2.25
50	20	1.25

Frequency Distribution

Price	Frequency	Relative Frequency
1.25	6	0.12
2.25	20	0.40
3.00	17	0.34
3.50	7	0.14
Total	50	1.00

Relative frequencies are fairly close to the given probabilities.

Measures

Mean = 2.56

Standard deviation = 0.6656

Now, to get the solution to the asked question,

vide(1) under Back-up Theory,

95% Confidence Interval is:

2.56 ±0.1890

Lower bound: $2.37; Upper bound: $2.75 ANSWER

Details of calculations

Given	α =	0.05
	n =	50
	Xbar =	2.56
	s =	0.6654
	tα/2 =	2.009575

95% CI for μ: 2.56 ± 0.18910458

Lower Bound =	2.3709
Upper Bound =	2.7491

DONE