Question

X = [0.01, 0.36, 0.97, 0.44, 0.86, 0.49, 0.02, 0.19, 0.54, 0.63, 0.69, 0.27, 0.21, 0.55, 0.54, 0.1 , 0.02, 0.49 , 0.42, 0.79], is a list of 20 random numbers generated from a pseudo-random number generator G.

a. What is a uniformity test?

b. Using Chi-square Goodness-of-fit test with 4 bins, test whether X belongs to U(0, 1) based on 5% significant level. 

c. Can we conclude G generates truly random numbers following the distribution U(0, 1)?

Answer 1

a.

$\chi^2$ squared test for uniformity can be formulated by binning the ranks 0: M into J bins and testing that the bins all have roughly the expected number of draws in them.

If b_j is the number of ranks that fall into bin j and e_j is the number of ranks expected to fall into

bin j, the test statistic is

$\chi^2=\sum \frac{(b_j-e_j)^2}{e_j}$

b.

$H_0:$ X belongs to U(0, 1)

$H_a:$ X does not belong to U(0, 1)

$X=[0.01, 0.02, 0.02, 0.1, 0.19, 0.21, 0.27, 0.36, 0.42, 0.44, 0.49, 0.49,$

$0.54, 0.54, 0.55, 0.63, 0.69, 0.79, 0.86, 0.97]$

4 bins:

$[0,0.25],[0.25,0.5],[0.5,0.75],[0.75,1]$

$\chi^2=\sum \frac{(b_j-e_j)^2}{e_j}$

expected number of values in each interval is

$e_j=20/4=5$

then:

$\chi^2=\frac{(6-5)^2+(6-5)^2+(5-5)^2+(3-5)^2}{5}=6/5=1.2$

$df=4-1=3$

critical value:

$\chi^2_{crit}=0.216$

c.

Since $\chi^2>\chi^2_{crit}$ we accept the null hypothesis. X belongs to U(0, 1)