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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.

\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & With Seaweed & Without Seaweed & Total \\
\hline Farm A & 50 & 36 & 86 \\
\hline Farm B & 74 & 40 & 114 \\
\hline Total & 124 & 76 & 200 \\
\hline
\end{tabular}

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 [tex][tex]$P(\text{farm A}|\text{with seaweed}) \neq P(\text{farm A})$[/tex][/tex].

B. A cow having seaweed in its feed and being from farm [tex][tex]$A$[/tex][/tex] are independent because [tex][tex]$P(\text{with seaweed}|\text{farm A}) = P(\text{with seaweed})$[/tex][/tex].

C. A cow being from farm [tex][tex]$B$[/tex][/tex] and not having seaweed in its feed are dependent because [tex][tex]$P(\text{farm B}|\text{without seaweed}) \neq P(\text{without seaweed})$[/tex][/tex].

D. A cow not having seaweed in its feed and being from farm [tex][tex]$B$[/tex][/tex] are independent because [tex][tex]$P(\text{without seaweed}|\text{farm B}) = P(\text{farm B})$[/tex][/tex].


Sagot :

To determine which statement is true based on the data provided in the table, we will assess the independence or dependence of events by examining their probabilities and conditional probabilities.

Given:
- Total number of cows: 200
- Cows with seaweed: 124
- Cows without seaweed: 76
- Cows from Farm A: 86
- Cows from Farm B: 114

Step-by-step solution:

1. Calculate the probabilities for Farm A and Farm B:
- [tex]\( P(\text{Farm A}) = \frac{86}{200} = 0.43 \)[/tex]
- [tex]\( P(\text{Farm B}) = \frac{114}{200} = 0.57 \)[/tex]

2. Calculate the probabilities for having feed with and without seaweed:
- [tex]\( P(\text{With Seaweed}) = \frac{124}{200} = 0.62 \)[/tex]
- [tex]\( P(\text{Without Seaweed}) = \frac{76}{200} = 0.38 \)[/tex]

3. Calculate the joint probabilities:
- [tex]\( P(\text{Farm A and With Seaweed}) = \frac{50}{200} = 0.25 \)[/tex]
- [tex]\( P(\text{Farm B and Without Seaweed}) = \frac{40}{200} = 0.2 \)[/tex]

4. Calculate the conditional probabilities:
- [tex]\( P(\text{With Seaweed | Farm A}) = \frac{50}{86} \approx 0.5814 \)[/tex]
- [tex]\( P(\text{Without Seaweed | Farm B}) = \frac{40}{114} \approx 0.3509 \)[/tex]

5. Check for independence:
- For independence of Farm A and having seaweed in its feed:
- Check if [tex]\( P(\text{Farm A and With Seaweed}) = P(\text{Farm A}) \times P(\text{With Seaweed}) \)[/tex]
- [tex]\( P(\text{Farm A}) \times P(\text{With Seaweed}) = 0.43 \times 0.62 = 0.266 \neq 0.25 \)[/tex]
- Thus, Farm A and having seaweed are dependent.
- For independence of having seaweed given Farm A:
- Check if [tex]\( P(\text{With Seaweed | Farm A}) = P(\text{With Seaweed}) \)[/tex]
- [tex]\( P(\text{With Seaweed | Farm A}) = 0.5814 \neq 0.62 \)[/tex]
- Thus, having seaweed and being from Farm A are dependent.
- For independence of Farm B and not having seaweed:
- Check if [tex]\( P(\text{Farm B and Without Seaweed}) = P(\text{Farm B}) \times P(\text{Without Seaweed}) \)[/tex]
- [tex]\( P(\text{Farm B}) \times P(\text{Without Seaweed}) = 0.57 \times 0.38 = 0.2166 \neq 0.2 \)[/tex]
- Thus, Farm B and not having seaweed are dependent.
- For independence of not having seaweed given Farm B:
- Check if [tex]\( P(\text{Without Seaweed | Farm B}) = P(\text{Without Seaweed}) \)[/tex]
- [tex]\( P(\text{Without Seaweed | Farm B}) = 0.3509 \neq 0.38 \)[/tex]
- Thus, not having seaweed and being from Farm B are dependent.

Answer Analysis:

- Statement A: A cow being from Farm A and having seaweed in its feed are dependent because [tex]\( P(\text{Farm A and With Seaweed}) \neq P(\text{Farm A}) \times P(\text{With Seaweed}) \)[/tex]. True
- Statement B: A cow having seaweed in its feed and being from Farm A are independent because [tex]\( P(\text{With Seaweed | Farm A}) = P(\text{With Seaweed}) \)[/tex]. False
- Statement C: A cow being from Farm B and not having seaweed in its feed are dependent because [tex]\( P(\text{Farm B and Without Seaweed}) \neq P(\text{Farm B}) \times P(\text{Without Seaweed}) \)[/tex]. True
- Statement D: A cow not having seaweed in its feed and being from Farm B are independent because [tex]\( P(\text{Without Seaweed | Farm B}) = P(\text{Without Seaweed}) \)[/tex]. False

Therefore, the correct statements are A and C. However, since we need to choose one statement, we select:
A. A cow being from Farm A and having seaweed in its feed are dependent because [tex]\( P(\text{farm A | with seaweed}) \neq P(\text{farm A}) \)[/tex].