Question

. A consumer group wants to test the effectiveness of a new "organic" laundry detergent and make recommendations to customers about how to best use the product. They intentionally get grass stains on 30 white T-shirts in order to see how well the detergent will clean them. They want to try the detergent in cold water and in hot water on both the "regular" and "delicates" wash cycles. Design an appropriate experiment, indicating the number of factors, levels, and treatments. Explain the role of randomization in your experiment.

Short answer

The experiment has 2 factors (Water Temperature, Wash Cycle), each with 2 levels, creating 4 treatments. Randomization distributes T-shirts to treatments fairly.

Step-by-step solution

Identify the Factors

The factors in the experiment are the variables that will be altered to observe their effect on the outcome. In this case, the factors are "Water Temperature" and "Wash Cycle Type". Each of these factors will have different levels that will be tested.

Determine the Levels

Each factor identified in Step 1 has specific levels at which they can be tested. 1. Water Temperature has 2 levels: cold water and hot water. 2. Wash Cycle Type has 2 levels: regular and delicates.

Calculate the Number of Treatments

To find the number of treatments, multiply the number of levels of each factor. Since "Water Temperature" has 2 levels and "Wash Cycle Type" also has 2 levels, there are 2 x 2 = 4 treatments in total. These treatments are: 1. Cold water with regular cycle 2. Cold water with delicates cycle 3. Hot water with regular cycle 4. Hot water with delicates cycle

Explain Role of Randomization

Randomization is crucial to eliminate bias and ensure that the results of the experiment are due to the factors being tested and not some uncontrolled variable. In this experiment, randomization can be done by randomly assigning each of the 30 T-shirts to one of the 4 treatments. This ensures a fair distribution of factors across the different conditions.

Design the Experiment

Divide the 30 T-shirts such that each treatment receives an equal number of T-shirts. Since there are 4 treatments, each treatment would get 7 or 8 T-shirts, which can be roughly calculated by dividing 30 by 4. Randomly assign T-shirts to each treatment to ensure variability is evenly distributed.

Key concepts

Randomization

Randomization is a key component in designing reliable experiments. It involves randomly distributing the experimental units across various treatments to ensure that individual differences do not skew the results of the study.

In the context of testing a new organic laundry detergent, randomization means that each of the 30 T-shirts should be randomly assigned to one of the four treatments, which combine different water temperatures and wash cycles.

• This helps to avoid bias, where some shirts might inadvertently end up in the same treatment due to non-random assignment.
• Randomization contributes to the credibility and generalizability of the experiment's results by ensuring that the effect observed is truly due to the treatments rather than some unmeasured variables.

When each shirt has an equal chance of being selected for any treatment, the outcome provides a clearer picture of the detergent’s effectiveness under various conditions.

Factors and Levels

To design an experiment, you need to clearly define what aspects you will be varying—these are called factors. Factors describe the independent variables that you want to test in your experiment.

In the laundry detergent example, the primary factors are "Water Temperature" and "Wash Cycle Type." These determine how the experiment will be structured:

• **Water Temperature:** This factor has two levels, 'cold' and 'hot', which means the T-shirts will be washed with either cold or hot water.
• **Wash Cycle Type:** This factor also has two levels, 'regular' and 'delicates', indicating the washing cycle used during the test.

By identifying and setting specific levels for each factor, the experiment can be structured to assess how these variables influence the outcome. These levels are combined to form the different treatments that the T-shirts will undergo.

Controlled Experiment

A controlled experiment ensures that the only variables affecting the outcome are the ones being tested, helping to establish a cause-and-effect relationship.

In this laundry detergent testing, the "control" refers to how other potential variables—outside of water temperature and wash cycle—are kept constant.

• This typically involves using the same type of T-shirts, similar stain conditions, and consistent washing procedures apart from the factors being tested.
• It is done to negate the impact of extraneous variables that could otherwise introduce error into the results. By controlling these unwanted variables, it ensures that any difference in the effectiveness of stain removal can be attributed to the detergent and its usage conditions rather than other factors.

For instance, if some T-shirts are washed for a longer time than others, it could affect the results, making it hard to determine the true effectiveness of each treatment. Keeping everything else the same is vital for a reliable experiment.