A public health researcher is investigating the effectiveness of fluoride in preventing cavities. Two communities are studied, one with a fluoridiated water supply and the second with non-fluoridated water. A random sample of n1 = 400 and n2 = 900 residents are obtained from these two communities, respectively. It was found that yı = 348 residents in the first community and y2 = 819 residents in the second community had cavities. Consider modeling these counts by two binomial distributions with the probability of success (presence of cavities) as T1 and 72 for communities with fluoridated and non-fluoridated water supplies, respectively. (i.e. consider the random variables Yı ~ Bin(n1, T1) and Y2 ~ Bin(n2, T2).) a. Let P = Yı/n1 and P2 = Y2/n2 be random variables denoting the proportion of sampled residents with cavities in each population. Approximate the sampling distributions of P1 and P2. b. Based on your answer from (a), approximate the sampling distribution for the difference in sample proportions, D = P1 – P2. (Hint: think of this as a sum of normal random variables.) c. Suppose we use the statistic D as an estimator for the true difference 71 – 72. Calculate the bias and standard error of this estimator. (Note, these may be a function of T1, T2, N1, and n2.)

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.