## Three Building Block Designs Completely Randomized Design

One of the simplest experimental designs is the randomization and analysis plan that is used with a t statistic for independent samples. Consider an experiment to compare the effectiveness of two diets for obese teenagers. The independent variable is the two kinds of diets; the dependent variable is the amount of weight loss two months after going on a diet. For notational convenience, the two diets are called treatment A. The levels of treatment A corresponding to the specific diets are denoted by the lowercase letter a and a subscript: a1 denotes one diet and a2 denotes the other. A particular but unspecified level of treatment A is denoted by a, where j ranges over the values 1 and 2. The amount of weight loss in pounds 2 months after participant i went on diet j is denoted by Yij.

The null and alternative hypotheses for the weight-loss experiment are, respectively,

where and denote the mean weight loss of the respective populations. Assume that 30 girls who want to lose weight are available to participate in the experiment. The researcher assigns n = 15 girls to each of the p = 2 diets so that each of the (np)!/(n!)p = 155,117,520 possible assignments has the same probability. This is accomplished by numbering the girls from 1 to 30 and drawing numbers from a random numbers table. The first 15 numbers drawn between 1 and 30 are assigned to treatment level a1; the remaining 15 numbers are assigned to a2. The layout for this experiment is shown in Figure 1.1. The girls who were assigned to treatment level a1 are called Group1; those assigned to treatment level a2 are called Group2. The mean weight losses of the two groups of girls are denoted by Y 4 and Y 2.

The t independent-samples design involves randomly assigning participants to two levels of a treatment. A completely randomized design, which is described next, extends this design strategy to two or more treatment levels. The completely randomized design is denoted by the letters CR-p, where CR stands for "completely randomized" and p is the number of levels of the treatment.

Again, consider the weight-loss experiment and suppose that the researcher wants to evaluate the effectiveness of

Group j

Group 2

Participanti |
ai |

Participant |
a\ |

Participanti5 |
«1 |

Participant^ |
«2 |

Participant^ |
Ol |

Participant3o |
ai |

Figure 1.1 Layout for a t independent-samples design. Thirty girls are randomly assigned to two levels of treatment A with the restriction that 15 girls are assigned to each level. The mean weight loss in pounds for the girls in treatment levels a1 and a2 is denoted by Y 1 and Y 2, respectively.

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