A dairy farmer thinks that the average weight gain of his cows depends on two factors: the type of grain that they are fed and the type of grass that they are fed. The dairy farmer has four different types of grain from which to choose and three different types of grass from which to choose. He would like to determine if there is a particular combination of grain and grass that would lead to the greatest weight gain on average for his cows. He randomly selects three one-year-old cows and assigns them to each of the possible combinations of grain and grass. After one year he records the weight gain for each cow (in pounds) with the following results. Is there sufficient evidence to conclude that there is a significant difference in the average weight gains among the cows for the different types of grain? Cow Weight Gain (Pounds) Grass A Grass B Grass B Grain A 359359 327327 232232 277277 250250 163163 191191 304304 216216 Grain B 331331 348348 176176 318318 205205 311311 314314 356356 253253 Grain C 310310 256256 172172 299299 238238 247247 212212 285285 245245 Grain D 257257 208208 206206 348348 325325 338338 208208 263263 302302 ANOVA Source of Variation SSSS dfdf MSMS Grain 8044.55568044.5556 33 2681.51852681.5185 Grass 15957.388915957.3889 22 7978.69457978.6945 Interaction 11508.611111508.6111 66 1918.10191918.1019 Within 80417.333380417.3333 2424 3350.72223350.7222 Total 115927.8889115927.8889 3535 Step 2 of 2 : Make the decision to reject or fail to reject the null hypothesis of equal average weight gains among the cows for the different types of grain and state the conclusion in terms of the original problem. Be sure to test for interaction first. Use α=0.01

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 28EQ

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Question

A dairy farmer thinks that the average weight gain of his cows depends on two factors: the type of grain that they are fed and the type of grass that they are fed. The dairy farmer has four different types of grain from which to choose and three different types of grass from which to choose. He would like to determine if there is a particular combination of grain and grass that would lead to the greatest weight gain on average for his cows. He randomly selects three one-year-old cows and assigns them to each of the possible combinations of grain and grass. After one year he records the weight gain for each cow (in pounds) with the following results. Is there sufficient evidence to conclude that there is a significant difference in the average weight gains among the cows for the different types of grain?

Cow Weight Gain (Pounds)

	Grass A	Grass B	Grass B
Grain A	359359	327327	232232
277277	250250	163163
191191	304304	216216
Grain B	331331	348348	176176
318318	205205	311311
314314	356356	253253
Grain C	310310	256256	172172
299299	238238	247247
212212	285285	245245
Grain D	257257	208208	206206
348348	325325	338338
208208	263263	302302

ANOVA

Source of Variation	SSSS	dfdf	MSMS
Grain	8044.55568044.5556	33	2681.51852681.5185
Grass	15957.388915957.3889	22	7978.69457978.6945
Interaction	11508.611111508.6111	66	1918.10191918.1019
Within	80417.333380417.3333	2424	3350.72223350.7222
Total	115927.8889115927.8889	3535

Step 2 of 2 :

Make the decision to reject or fail to reject the null hypothesis of equal average weight gains among the cows for the different types of grain and state the conclusion in terms of the original problem. Be sure to test for interaction first. Use α=0.01

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