Question

Comparing weight-loss treatments $A$total of $20$overweight females have agreed to participate in a study of the effectiveness of four weight-loss treatments: $A,B,C,andD$. The researcher first calculates how overweight each subject is by comparing the subject’s actual weight with her “ideal” weight. The subjects and their excess weights in pounds are as follows

The response variable is the weight lost after $8$weeks of treatment.

a. If the $5$most overweight women were assigned Treatment $A$, the next $5$most overweight women were assigned Treatment $B$, and so on, then the amount overweight and type of treatment would be confounded. Explain this statement carefully.

b. Describe how to randomly assign the women to treatments using a randomized block design. Use blocks of size $4$formed by the amount overweight.

c. Use technology or Table $D$to carry out the random assignment

Short answer

(a) The difference in the treatment groups is due to the weight of subjects rather than the treatment itself.

(b) One person in each block is randomly assigned to treatment $A$, one of the remaining individuals in each block is randomly allocated to treatment $B$, one of the remaining people in each block is randomly assigned to treatment $C$, and the other people are all assigned to treatment $D$.

(c) The random assignments can vary according to the treatment.

Step-by-step solution

Part (a) Step 1: Given information

We need to explain the statement "If the five most obese women were given Treatment $A$, the next most obese women were given Treatment $B$, and so on, the level of obesity and the type of treatment would be muddled".

Part (a) Step 2: Explanation

We know that

• When the effects of two variables on the same response variable cannot be distinguished, they are said to be confused.
• Because the five most overweight women are assigned to treatment $A$, the next five most overweight women are assigned to treatment $B$, and so on, we cannot conclude that the variations in the treatments are due to the treatments (for treatment $C$and treatment $D$).

The discrepancies between the treatment groups could therefore be attributed to the participants' weight rather than the treatment itself, confounding the degree of overweight and the type of treatment.

Part (b) Step 1: Given information

We need to find the reason for randomly assigning the women to treatments using a randomized block design.

Part (b) Step 2: Explanation

We know that

One person in each block is randomly assigned to treatment $A$, one of the remaining individuals in each block is randomly allocated to treatment $B$, one of the remaining people in each block is randomly assigned to treatment $C$, and the other people are all assigned to treatment $D$.

Compare the weight loss for the different blocks and treatments after $8$weeks of treatment.

Part (c) Step 1: Given information

We need to assign the random values.

Part (c) Step 2: Explantion

Let us begin from row $115$in table $D$i.e.

$61041776849432224709736981452631893325921445926056$

Select the first digit and if digit is between one and four then it is assigned to treatment $A$otherwise ignore the digit and select next one. repeat this for all the treatments and blocks.

Therefore, random assignment may be,

Treatment $A$is Williams, Kendall, Brunk, Brown, Tran.

Treatment $B$is Moses, Smith, Loren, Jackson, Birnbaum.

Treatment $C$is Hernandez, Mann, Obrach, Stall, Wilansky.

Treatment $D$is Deng, Santiago, Rodriguez, Cruz, Nevesky.