3.3 Generate 1000 uniform (0,I) numbers using a spreadsheet. (a) Use the Kolmogo
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3.3 Generate 1000 uniform (0,I) numbers using a spreadsheet. (a) Use the Kolmogorov-Smirnov test with alpha = 0.05 to test if the hypothesis that the numbers are uniformly distributed on the (0,1) interval can be rejected. (b) Use a chi-square goodness-of-fit test with 10 intervals and alpha = 0.05 to test if the hypothesis that the numbers are uniformly distributed on the (0,1) interval can be rejected. (c) Test the hypothesis that the numbers are uniformly distributed within the unit square, {(x,y) : x E (0,1 ), y E (0,1)) using the 2D chi-squared test at a 95% con- (d) Test the hypothesis that the numbers have a lag-1 correlation of zero. Make an (e) If you have access to a suitable statistical package, perform a runs above the mean fidence level. Use 10 intervals for each of the dimensions. autocorrelation plot of the numbers. test to check the randomness of the numbers. Use the nonparametric runs test functionality of your statistical software to perform this test. (t) What is your conclusion conceming the suitability of your spreadsheet's random number generator?Explanation / Answer
Descriptive Statistics
N
Mean
Std. Deviation
Minimum
Maximum
VAR00001
1000
.5039
.29785
.00
1.00
One-Sample Kolmogorov-Smirnov Test
VAR00001
N
1000
Uniform Parametersa,,b
Minimum
.00
Maximum
1.00
Most Extreme Differences
Absolute
.030
Positive
.015
Negative
-.030
Kolmogorov-Smirnov Z
.941
Asymp. Sig. (2-tailed)
.338
a. Test distribution is Uniform.
b. Calculated from data.
Since KS Z = 0.941 and Significance level is 0.338 > 0.05 it means that data is U(0,1)
Lower Limit
Upper Limit
Cum. Freq
Frequncy
Exp Freq
0
0.1
110
110
100
0.1
0.2
209
99
100
0.2
0.3
302
93
100
0.3
0.4
405
103
100
0.4
0.5
499
94
100
0.5
0.6
583
84
100
0.6
0.7
678
95
100
0.7
0.8
775
97
100
0.8
0.9
892
117
100
0.9
1
1000
108
100
Chi-Square Value
8.38
P-value
0.496351
P-value > 0.05 means Data supports U(0,1)
X * Y Crosstabulation
Count
Y
Total
1
2
3
4
5
6
7
8
9
X
2
19
9
9
11
4
12
13
8
14
99
3
20
13
10
9
6
8
8
12
7
93
4
24
8
6
14
11
9
6
16
9
103
5
23
15
7
9
6
14
11
5
4
94
6
16
9
8
9
11
6
11
7
7
84
7
21
7
7
9
7
16
13
8
7
95
8
21
10
14
9
12
11
4
10
6
97
9
17
12
13
10
14
13
8
16
14
117
10
38
24
22
20
17
32
25
22
18
218
Total
199
107
96
100
88
121
99
104
86
1000
Chi-Square Tests
Value
df
Asymp. Sig. (2-sided)
Pearson Chi-Square
60.223a
64
.611
Likelihood Ratio
61.380
64
.570
Linear-by-Linear Association
.465
1
.495
N of Valid Cases
1000
a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 7.22.
As P value = 0.611 > 0.05 Data supports U(0,1)
v
Runs Test
VAR00001
Test Valuea
.50
Cases < Test Value
500
Cases >= Test Value
500
Total Cases
1000
Number of Runs
514
Z
.823
Asymp. Sig. (2-tailed)
.411
a. Median
Runs Test 2
VAR00001
Test Valuea
.5039
Cases < Test Value
503
Cases >= Test Value
497
Total Cases
1000
Number of Runs
516
Z
.950
Asymp. Sig. (2-tailed)
.342
a. Mean
Descriptive Statistics
N
Mean
Std. Deviation
Minimum
Maximum
VAR00001
1000
.5039
.29785
.00
1.00
One-Sample Kolmogorov-Smirnov Test
VAR00001
N
1000
Uniform Parametersa,,b
Minimum
.00
Maximum
1.00
Most Extreme Differences
Absolute
.030
Positive
.015
Negative
-.030
Kolmogorov-Smirnov Z
.941
Asymp. Sig. (2-tailed)
.338
a. Test distribution is Uniform.
b. Calculated from data.
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