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 being from farm a and having seaweed in its feed are dependent because p(farm a|with seaweed) ≠ p(farm a)
b. a cow having seaweed in its feed and being from farm a are independent because p(with seaweed|farm a) = p(with seaweed)
c. a cow not having seaweed in its feed and being from farm b are independent because p(without seaweed|farm b) = p(farm b)
d. a cow being from farm b and not having seaweed in its feed are dependent because

	farm a	farm b	total
without seaweed	30	40	76
total	80	114	200

Explanation:

Step1: Recall independence formula

Two events \(A\) and \(B\) are independent if \(P(A|B)=P(A)\) or \(P(A\cap B) = P(A)\times P(B)\). In the context of the two - way table, let event \(A\) be the cow being from a particular farm and event \(B\) be the cow having seaweed in its feed.

Step2: Calculate conditional and marginal probabilities

For example, \(P(\text{farm A})=\frac{86}{200}=0.43\), \(P(\text{with seaweed})=\frac{124}{200} = 0.62\), \(P(\text{farm A and with seaweed})=\frac{50}{200}=0.25\). And \(P(\text{farm A})\times P(\text{with seaweed})=0.43\times0.62 = 0.2666
eq0.25\). In general, we check the relationships for all combinations. If \(P(\text{farm A}|\text{with seaweed})=\frac{P(\text{farm A and with seaweed})}{P(\text{with seaweed})}=\frac{50/200}{124/200}=\frac{50}{124}\approx0.403\) and \(P(\text{farm A})=\frac{86}{200} = 0.43\), they are not equal.
We know that if \(P(\text{farm A}|\text{with seaweed})
eq P(\text{farm A})\), the events are dependent.

Answer:

A. A cow being from farm A and having seaweed in its feed are dependent because \(P(\text{farm A}|\text{with seaweed})
eq P(\text{farm A})\)