# Homework Assignment for Lesson 3

### Homework Assignment

**1. **Consider a completely randomized design with observations on three treatments coded 1,2,3. For the one-way ANOVA model, determine which of the following are estimable. For those that are estimable, write out the estimable function as $\sum_{i=1}^3 b_i (\mu+\tau_i)$ and clearly state $b_1,b_2,b_3$. Finally, for those that are estimable, state the least squares estimator.

- $\tau_1+\tau_2-2\tau_3$
- $\mu+\tau_3$
- $\tau_1-\tau_2-\tau_3$
- $\mu+(\tau_1+\tau_2+\tau_3)/3$

**2.** Recall the soap experiment from Homework 1. Look back at Homework 1 for an explanation of the experiment. The data are the weight lost over 24 hours by different types of soap.

Cube |
Regular |
Deodorant |
Moisturizing |

1 | -0.30 | 2.63 | 1.86 |

2 | -0.10 | 2.61 | 2.03 |

3 | -0.14 | 2.41 | 2.26 |

4 | 0.40 | 3.15 | 1.82 |

- Write out the one-way ANOVA model for this experiment.
- By hand or calculator (without using SAS), obtain the LS estimate for the mean weight lost by a cube of deodorant soap. Show all calculations.
- Consider estimating the difference in weight loss between regular soap and any other type of soap. That is, consider estimating $\tau_{regular}-(\tau_{deodorant}+\tau_{moisturizing})/2$.Show that this is estimable, and find the LS estimate by hand or calculator. Show all calculations.
- Now use SAS to obtain the LS estimates in parts (b) and (c). Include your SAS code and the relevant output in your homework.

**3. Pedestrian light experiment** (Larry Lesher, 1985)

This experiment questions whether pushing a certain pedestrian light button had an effect on the wait time before the pedestrian light showed “walk.” The treatment factor of interest was the number of pushes of the button, and 32 observations were taken with a mix of 0, 1, 2, and 3 pushes of the button. The waiting times for the “walk” sign are shown in the following table, with $r_0= 7,\ r_1= r_2= 10,\ r_3= 5$ (where the levels of the treatment factor are coded as 0, 1, 2, 3 for simplicity).

0 |
1 |
2 |
3 |

38.14 | 38.28 | 38.17 | 38.14 |

38.20 | 38.17 | 38.13 | 38.30 |

38.31 | 38.08 | 38.16 | 38.21 |

38.14 | 38.25 | 38.30 | 38.04 |

38.29 | 38.18 | 38.34 | 38.37 |

38.17 | 38.03 | 38.34 | |

38.20 | 37.95 | 38.17 | |

38.26 | 38.18 | ||

38.30 | 38.09 | ||

38.21 | 38.06 |

- Plot the waiting times against the number of pushes of the button. What does the plot show?
- Write out the one-way ANOVA model for this experiment.
- Use SAS to estimate the mean waiting time for each number of pushes.
- Show that the contrast $\tau_0-(\tau_1+\tau_2+\tau_3)/3$ is estimable, and use SAS to find it's LS estimate. This contrast compares the effect of no pushes of the button with the average effect of pushing the button once, twice, or three times.