# Homework Assignment for Lesson 4

Homework Assignment

**1. **Show that \(SSTOT / \sigma^2 \sim \chi_{n-1}^{2}\) by using an argument similar to that used in the lectures to find distributions for \(SSE / \sigma^2\) and \(SST/ \sigma^2\).

**2. **A test statistic that could be used to test for a significant pairwise di↵erences between the i-th and j-th treatments is

\[D_{ij}^{*}=\frac{(\bar{Y}_{i.} − \bar{Y}_{j.})^2}{SSE/(n − v)}\]

The null hypothesis is \(H_0: \tau_i = \tau_j\) . Find the distribution of \(D_{ij}^{*}under the null hypothesis.

(For your information, SAS uses the square root of this test statistic to get the* p*-values reported in the LSMEANS output)

**3. **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 |

(a) Construct the ANOVA table for this experiment by hand. Show the calculations needed to construct the quantities in the ANOVA table.

(b) Test the null hypothesis that there is no difference in mean weight lost between different soap types. Report the test statistic, the *p*-value of the statistic under the null hypothesis, and interpret the result of the test. You may use SAS or an online *p*-value calculator (like http://graphpad.com/quickcalcs/PValue1.cfm ) to compute the *p*-value. Provide the SAS code used, with output, or clearly describe the online *p*-value calculator you used.

**4. Pedestrian Light Experiment **(Larry Lesher, 1985)

Recall the pedestrian light experiment from Homework 4. 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).

1 |
2 |
3 |
4 |

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 |

Answer the question: “**Does pushing the button make the light change sooner?**”. Clearly state the null and alternative hypotheses, the model used, and all assumptions in the model. Obtain a test statistic (show all code and important SAS output), and interpret the results of the test.

**5. Hot Dogs **

A study was conducted to compare the calories and sodium in hot dogs made with different types of meat. Use following data in the file hotdog.txt in SAS, within a DATA statement.

(a) Read the data into SAS and plot calories as a response variable with the type of meat on the x-axis. Your plot could be either a boxplot or a plot with one dot for each hot dog.

(b) Answer the following question: “**Are there differences in the average calories of hot dogs made with different kinds of meat?**”. To answer this question, write down a statistical model (clearly state the response variable, treatment levels, number of replicates, ...), express the above question as a testable null hypothesis, and report the p-value of the test statistic under the null hypothesis. Your answer should include all SAS code used, and the important SAS output.

(c) Are there significant di↵erences in mean calories between Beef and Pork hot dogs? What about between Beef and Chicken hot dogs? What about between Pork and Chicken hot dogs?