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There are n cards serially numbered from 1 to n. The cards are shuffled and placed in a line one after the other on top of a table with faces up. The numbers on the faces are read from left to right. If there are consecutive numbers in increasing order of magnitude the corresponding cards are merged into one. After the merger, the cards are numbered serially from one to whatever the number of cards we now have. The cards are shuffled and placed in a line one after another on top of the table with faces up. The process continues until we have only one card left. In this paper, we develop a probabilistic recurrence relation approach to obtain the mean, variance, and distribution of the number of shuffles needed. A Markov chain formulation and its properties are discussed in the paper as well.

In this paper, the test for equality of several intraclass correlation coefficients is proposed when the sample size is small. First, we consider a case with two populations. The exact null distribution of the test statistic is derived. We use a Monte Carlo simulation study and find that the performance of our new test is superior to the tests that were proposed by Young and Bhandary (1998). Work is then extended to develop a test for three intraclass correlation coefficients. Monte Carlo simulation studies are also conducted to compare our new test and the test that was proposed by Bhandary and Alam (2000). Based on our simulation result, we recommend our new test in practice when the sample sizes are small.

Whittemore (1981) and Hsieh et al. (1998) have proposed different methods for determining sample size in the context of testing the significance of a slope parameter in logistic regression. Their sample size formulas have been incorporated in some statistical software packages. In this paper, we use a variation of Whittemore (1981) method to calculate sample size. We compare these three sample size formulas to assess closeness of the nominal and observed powers via simulations. These studies lead us to propose another method, which is a combination of Hsieh et al. (1998) method and the proposed variation of the Whittemore (1981) method for calculating sample size when the covariate is normally distributed. However, when the covariate has a Bernoulli distribution, sample size calculations widely diverge between the Hsieh et al. (1998) method and the proposed method. Interestingly, the proposed method gives a better account of power than the Hsieh et al. (1998) method.