# Homework Assignment for Lesson 8

### Homework Assignment

**1. Widgets (hypothetical "any-products")**

Download the "widgets.sas'' code from Angel. This code will input data into SAS for the following situation.

Suppose that you have a business that uses widgets as raw material for a product that you manufacture and sell. The widgets are currently purchased from several different suppliers. You notice that your machines are constantly breaking down, and find out that this is due to excessive variability in the sizes of the widgets. If you can make sure that your widgets are more uniform in size, then you can adjust your machines to allow for this.

The widget variability could be due to any of three reasons: (1) differences between suppliers, in which case you could restrict to one or two suppliers at a time, or else tell each of your factories to purchases widgets only from the supplier that is closest, (2) differences between average sizes for different batches of widgets from the same supplier. In that case, you can test one or two widgets from each box and return boxes with widgets whose average size is too large or too small, or (3) variability within each batch of widgets, in which case there may be no simple solution other either buying (or designing) more robust machines or else negotiating with your suppliers to provide more uniform widgets.

As a way to determine the source of the machine-damaging size variation in the widgets, you select 3 suppliers and order 4 boxes of widgets from each supplier at different times. You select 4 widgets from each box for testing, for a total of 3 × 4 × 4 = 48 widgets.

- Write out the 2-way nested model for this experiment.
- Explain why one factor is a "nested'' factor.
- Plot the data.
- Fit the model using PROC MIXED in SAS and examine the residuals. Transform the response if needed to address any problems with normality or constant error variance. If you transform the response, clearly show the residuals from the un-transformed response, and your best transformation, and describe why you chose the transformation you did.
- Identify a sum of squares that could be used to partition the variation in widget size into (1) variation due to differences between suppliers, (2) differences between average sizes for different batches of widgets from the same supplier, and (3) variability within each batch of widgets. Which of these sums of squares is the greatest for this dataset?
- Interpret the results of the hypothesis tests conducted by PROC MIXED in the context of this experiment.

**2. Nested vs. Crossed**

For the following descriptions of experiments, write out an appropriate model for the experiment. Clearly show which factors are nested and which are crossed. It may be helpful (but is not required) to draw a diagram of the experiment.

- An experiment is conducted in which 30 runners (15 men and 15 women) were randomly assigned to one of three physical training groups (endurance training, strength training, and cross training). The randomization is conducted so that 5 men and 5 women are assigned to each of the three training groups. The race times of the runners were recorded both before and after training for one month. The response variable of interest is the difference between before and after race times.
- An experiment is conducted to test how much time students from different majors spend studying. Three students are randomly selected from all students majoring in Statistics. Three separate students are randomly selected from all students majoring in Humanities. Each student recorded how much time they spent studying on three different days.
- An engineer wants to study variability in the strength of glass cathode supports made on production machines in the manufacturing plant. There are five production machines in the plant, and each machine has 4 components called ‘heads’ which produce the glass cathode supports. The engineer took 4 glass supports made from each head and tested their strength. Data collection of the 5 × 4 × 4 = 80 measurements was completely randomized.