Gap Test

• The gap test is used to determine the significance of the interval between recurrence of the same digit.

• A gap of length x occurs between the recurrence of some digit.

• See the example on page 313 where the digit 3 is underlined. There are a total of eighteen 3's in the list. Thus only 17 gaps can occur.

• The probability of a particular gap length can be determined by a Bernoulli trail.
  
  $$P ({\rm gap ~of} ~ n) = P(x \ne 3) P(x \ne 3) ... P(x \ne 3) P (x = 3)$$
  
  
  If we are only concerned with digits between 0 and 9, then
  
  $$P ({\rm gap ~of} ~ n) = 0.9^{n} 0.1$$
  
  

• The theoretical frequency distribution for randomly ordered digits is given by
  
  $$P (gap \le x) = F(x) = 0.1 \sum_{n=0}^x (0.9)^n = 1 - 0.9^{x+1}$$
  
  

• Steps involved in the test.

  Step 1.

  Specify the cdf for the theoretical frequency distribution given by Equation (8.14) based on the selected class interval width (See Table 8.6 for an example).

  Step 2.

  Arrange the observed sample of gaps in a cumulative distribution with these same classes.

  Step 3.

  Find D, the maximum deviation between F(x) and $S_N(x)$ as in Equation 8.3 (on page 299).

  Step 4.

  Determine the critical value, $D_\alpha$, from Table A.8 for the specified value of $\alpha$ and the sample size $N$.

  Step 5.

  If the calculated value of D is greater than the tabulated value of $D_\alpha$, the null hypothesis of independence is rejected.

• See the Example 8.13 on page 314