by P, is the same on every trial. ■ The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because: ■ The experiment consists of repeated trials. We flip a coin 2 times. ■ Each trial can result in just two possible outcomes - heads or tails. ■ The probability of success is constant - 0.5 on every trial. ■ The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials. Notation The following notation is helpful, when we talk about binomial probability. ■ x: The number of successes that result from the binomial experiment. ■ n: The number of trials in the binomial experiment. ■ P: The probability of success on an individual trial. ■ Q: The probability of failure on an individual trial. (This is equal to 1 - P.) ■ b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P. ■ nCr: The number of combinations of n things, taken r at a time. Binomial Distribution A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution (also known as a Bernoulli distribution). Suppose we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2. The binomial distribution is presented below.
A large z value would be the one above 2 02 below -2
Z = + or – 1.96 is equal to an alpha level of .05
Alpha is divided between the two tails 0.05 .025 each
Alpha is one side 0.05
Alpha is in one side 0.05
Z = + or – 1.96 is equal to an alpha level of .05
Given this background, let us now consider a trial in which groups receiving one of two therapies, therapy A and therapy B, are being compared. (Keep in mind the sampling of beads just discussed.) Before beginning our study, we can list the four possible study outcomes: It is possible that in reality there is no difference in efficacy between therapy A and therapy B (i.e., therapy A is no better and no worse than therapy B), and when we do our study we correctly conclude on the basis of our samples that the two groups do not differ. It is possible that in reality there is no difference in efficacy between therapy A and therapy B (i.e., therapy A is no better and no worse than therapy B), but in our study we found a difference between the groups and therefore concluded, on the basis of our samples, that there is a difference between the therapies. This conclusion, based on our samples, is in error. It is possible that in reality there is a difference between therapy A and therapy B, but when we examine the groups in our study we find no difference between them. We therefore conclude, on the basis of our samples, that there is no difference between therapy A and therapy B. This conclusion is in error. It is possible that in reality there is a difference between therapy A and therapy B, and when we examine the groups in our study we find that they differ. On the basis of these samples, we correctly conclude that therapy A differs from therapy B.
These four possibilities constitute the universe of outcomes after we complete our study. Let us look at these four possibilities as presented in a 2 × 2 table: Two columns represent reality-either therapy A differs from therapy B or therapy A does not differ from therapy B. The two rows represent our decision: We conclude either that they differ or that they do not differ. In this figure, the four possibilities that were just listed are represented as four cells in the 2 × 2 table. If there is no difference, and on the basis of the samples included in our study we conclude there is no difference, this is a correct decision (cell a). If there is a difference, a