Enter Game FIN - K.
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ind Statistics
\begin{tabular}{|l|l|l|l|l|}
\hline & x & Y & 2 & Total \\
\hline A & 10 & 80 & 61 & 151 \\
\hline B & 110 & 44 & 126 & 280 \\
\hline C & 60 & 59 & 110 & 229 \\
\hline Total & 180 & 183 & 297 & 660 \\
\hline
\end{tabular}
Which statement is true about whether Z and B are independent events?
A. Z and B are not independent events because [tex]P( Z \mid B ) \neq P( Z )[/tex].
B. Z and B are independent events because [tex]P( Z \mid B )=P( Z )[/tex].
C. Z and B are independent events because [tex]P( Z \mid B )=P(B)[/tex].
D. Z and B are not independent events because [tex]P( Z \mid B ) \neq P(B)[/tex].
Answer (2)
Calculate P ( Z ∣ B ) using the formula: P ( Z ∣ B ) = 280 126 = 0.45 .
Calculate P ( Z ) using the formula: P ( Z ) = 660 297 = 0.45 .
Compare P ( Z ∣ B ) and P ( Z ) .
Since P ( Z ∣ B ) = P ( Z ) , Z and B are independent events: P ( Z mi d B ) = P ( Z ) .
Explanation
Analyze the problem We are given a contingency table and asked to determine if events Z and B are independent. Two events are independent if and only if P ( Z ∣ B ) = P ( Z ) . We need to calculate these probabilities and compare them.
Calculate P(Z|B) First, we calculate P ( Z ∣ B ) , which is the probability of Z occurring given that B has occurred. This is calculated as: P ( Z ∣ B ) = P ( B ) P ( Z ∩ B ) = count of B count of Z and B From the table, the count of Z and B is 126, and the count of B is 280. Therefore, P ( Z ∣ B ) = 280 126 Calculating this value: P ( Z ∣ B ) = 0.45
Calculate P(Z) Next, we calculate P ( Z ) , which is the probability of Z occurring. This is calculated as: P ( Z ) = total count total count of Z From the table, the total count of Z is 297, and the total count is 660. Therefore, P ( Z ) = 660 297 Calculating this value: P ( Z ) = 0.45
Compare P(Z|B) and P(Z) Now, we compare P ( Z ∣ B ) and P ( Z ) . We found that: P ( Z ∣ B ) = 0.45 P ( Z ) = 0.45 Since P ( Z ∣ B ) = P ( Z ) , events Z and B are independent.
Conclusion Since P ( Z ∣ B ) = P ( Z ) , the correct statement is: Z and B are independent events because P ( Z ∣ B ) = P ( Z ) .
Examples
Understanding independence between events is crucial in many real-world scenarios. For example, in marketing, a company might want to know if purchasing a product (event B) is independent of seeing an advertisement (event Z). If they are independent, it means the advertisement isn't influencing the purchase, and the company might need to rethink its advertising strategy. In medical studies, researchers check if a disease (event B) is independent of a certain genetic marker (event Z) to understand potential genetic links. In finance, one might check if a stock's performance (event B) is independent of overall market trends (event Z) to assess its risk profile. These examples show how testing for independence helps in making informed decisions across various fields.
Events Z and B are independent because the probability of Z given B equals the probability of Z. Thus, the correct answer is option B: Z and B are independent events because P ( Z ∣ B ) = P ( Z ) .
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