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Asked Mar 7, 2019

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The Agricultural Experiment Station of a university tested two different herbicides and their effects on crop yield. From 90 acres set aside for the experiment, the station used herbicide 1 on a random sample of 30 acres and herbicide 2 on a second random sample of 30 acres; they used the remaining 30 acres as a control. At the end of the growing season, the yields (in bushels per acre) were as follows:

Herbicide 1

81.2 | 81.1 | 79.9 | 84.6 | 80.4 | 74.4 | 81.7 | 90.1 | 102.4 | 89.2 |

92.0 | 91.7 | 75.9 | 95.1 | 76.1 | 83.0 | 88.0 | 80.5 | 73.6 | 80.4 |

103.2 | 85.9 | 73.6 | 80.0 | 82.4 | 79.5 | 99.8 | 96.6 | 81.3 | 94.7 |

Herbicide 2

94.8 | 90.9 | 85.2 | 83.3 | 95.5 | 85.4 | 87.1 | 89.6 | 83.7 | 88.7 |

91.4 | 85.4 | 89.0 | 92.4 | 85.0 | 91.0 | 89.2 | 100.9 | 88.5 | 90.3 |

87.6 | 80.7 | 90.0 | 101.0 | 92.1 | 97.9 | 92.5 | 88.8 | 89.4 | 100.1 |

Control

94.7 | 79.5 | 91.4 | 82.6 | 96.9 | 85.4 | 80.8 | 90.6 | 88.6 | 80.8 |

78.1 | 82.5 | 93.5 | 83.1 | 90.5 | 89.2 | 82.0 | 84.1 | 90.1 | 84.5 |

81.2 | 92.4 | 90.5 | 82.0 | 106.6 | 96.9 | 76.1 | 101.8 | 77.5 | 88.8 |

a. Compare the mean yields of herbicide 1 and herbicide 2 to the control treatment. Use a = .05.

b. Should the procedure you used in part (a) be a one-sided or a two-sided procecedures?

c. Interpret your findings in part (a).

Step 1

**Part (a):**

In order to compare the mean yields of Herbicides 1 and 2 to the Control, first perform one-way ANOVA at level of significance *α* = 0.05. Then, if the ANOVA suggests significant difference between the 3 treatments, perform pairwise post-hoc tests, once between pairs of treatments (Herbicide 1, Control), once between pairs (Herbicide 2, Control) and once between pairs (Herbicide 1, Herbicide 2).

The test is conducted here by using EXCEL. The software procedure is given below:

- Enter the data for the three treatments in three columns (or rows) and in the first row of the three columns, give respective labels
**Herbicide 1**,**Herbicide 2**,**Control**. - Select
**Data > Data Analysis > Anova: Single Factor > OK**. - Enter
**Input Range**as**$A$1:$C$31**, select**Columns**in**Grouped By**, tick on**Labels in first row**, enter**Alpha**as**05**and click**OK**.

The output using EXCEL is as follows:

(Note the highlighted cells)

Step 2

**Post hoc tests:**

The *P*-value as highlighted (≈ 0.029) in Step 1 is less than *α* = 0.05. Thus, all the treatments are not equally effective and there exists significant differences in the mean yields of the 2 herbicides and the control.

For post-hoc tests, there may be various methods. Here, Bonferroni method is implemented, which performs two samples *t*-tests between pairs of treatments.

It is assumed that the variances are equal in the three populations.

The test is conducted here by using EXCEL. The software procedure is given below:

- Select
**Data > Data Analysis > t-Test: Two-Sample Assuming Equal Variances > OK**. - To compare between Herbicide 1 and Control, enter
**Variable 1 Range**as**$A$1:$A$31**and**Variable 2 Range**as**$C$1:$C$31**, enter Hypothesized Mean Difference as**0**, tick on**Labels**, enter**Alpha**as**05**and click**OK**. - To compare between Herbicide 2 and Control, enter
**Variable 1 Range**as**$B$1:$B$31**and**Variable 2 Range**as**$C$1:$C$31**, enter Hypothesized Mean Difference as**0**, tick on**Labels**, enter**Alpha**as**05**and click**OK**. - To compare between Herbicide 1 and Herbicide 2, enter
**Variable 1 Range**as**$A$1:$A$31**and**Variable 2 Range**as**$B$1:$B$31**, enter Hypothesized Mean Difference as**0**, tick on**Labels**, enter**Alpha**as**05**and click**OK**.

The outputs using EXCEL is as follows:

(Note the highlighted cells)

Step 3

**Conclusion for part (a):**

Note the green, purple and orange highlights. For the first two pairwise tests, the *P*-values (approximately 0.299 and 0.090) are greater than *α* = 0.05. Thus, mean yield of neither Herbicide 1, nor Herbicide 2 is significantly d...

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