Question

a university is researching the impact of including seaweed in cattle feed. they assign feed with and without seaweed to be fed to cattle at two different dairy farms. the two - way table shows randomly collected data on 200 dairy cows from the two farms about whether or not their feed includes seaweed. based on the data in the table, which statement is true? a. a cow having seaweed in its feed and being from farm a are independent because p(with seaweed|farm a)=p(with seaweed). b. a cow being from farm b and not having seaweed in its feed are dependent because p(farm b|without seaweed)≠p(without seaweed). c. a cow being from farm a and having seaweed in its feed are dependent because p(farm a|with seaweed)≠p(farm a). d. a cow not having seaweed in its feed and being from farm b are independent because p(without seaweed|farm b)=p(farm b).

Explanation:

Step1: Recall the definition of independence

Two events are independent if \(P(A|B)=P(A)\). Let event \(A\) be having sea - weed in feed and event \(B\) be being from farm \(A\).

Step2: Calculate \(P(\text{with seaweed})\)

\(P(\text{with seaweed})=\frac{124}{200}=0.62\)

Step3: Calculate \(P(\text{with seaweed}|\text{farm }A)\)

\(P(\text{with seaweed}|\text{farm }A)=\frac{50}{86}\approx0.581\)
Since \(P(\text{with seaweed}|\text{farm }A)
eq P(\text{with seaweed})\), having seaweed in feed and being from farm \(A\) are dependent.
Let's check the general concept for conditional probability and independence for all options.
For two events \(X\) and \(Y\), independence means \(P(X|Y) = P(X)\) and \(P(Y|X)=P(Y)\).
For option A:
Let \(X\) be having seaweed in feed and \(Y\) be being from farm \(A\).
\(P(X)=\frac{124}{200}\), \(P(X|Y)=\frac{50}{86}\), \(P(X)
eq P(X|Y)\), so they are dependent.
For option B:
Let \(X\) be being from farm \(B\) and \(Y\) be not having seaweed in feed.
\(P(X)=\frac{114}{200}\), \(P(X|Y)=\frac{40}{76}\), \(P(X)
eq P(X|Y)\), so they are dependent.
For option C:
Let \(X\) be being from farm \(A\) and \(Y\) be having seaweed in feed.
\(P(X)=\frac{86}{200}\), \(P(X|Y)=\frac{50}{124}\), \(P(X)
eq P(X|Y)\), so they are dependent.
For option D:
Let \(X\) be not having seaweed in feed and \(Y\) be being from farm \(B\).
\(P(X)=\frac{76}{200}\), \(P(X|Y)=\frac{40}{114}\), \(P(X)
eq P(X|Y)\), so they are dependent.

The correct answer is based on the fact that for two events \(A\) and \(B\) to be independent \(P(A|B) = P(A)\) and \(P(B|A)=P(B)\). In the context of the table, we calculate the relevant conditional and marginal probabilities.

Answer:

B. A cow being from farm B and not having seaweed in its feed are dependent because \(P(\text{farm B}|\text{without seaweed})
eq P(\text{farm B})\)