\begin{tabular}{|l|c|c|c|}
\hline & Brand A & Brand B & Total \\
\hline Texas & 80 & 45 & 125 \\
\hline California & 96 & 54 & 150 \\
\hline Total & 176 & 99 & 275 \\
\hline
\end{tabular}

A person is randomly selected from those tested. Are being from Texas and preferring Brand A independent events? Why or why not?

A. No, they are not independent because [tex]$P(\text{Texas}) \approx 0.45$[/tex] and [tex]$P(\text{Texas} \mid \text{Brand A}) \approx 0.45$[/tex].

B. Yes, they are independent because [tex]$P(\text{Texas}) \approx 0.45$[/tex] and [tex]$P(\text{Texas} \mid \text{Brand A}) \approx 0.45$[/tex].

C. No, they are not independent because [tex]$P(\text{Texas}) \approx 0.45$[/tex] and [tex]$P(\text{Texas} \mid \text{Brand A}) \approx 0.64$[/tex].

D. Yes, they are independent because [tex]$P(\text{Texas}) \approx 0.45$[/tex] and [tex]$P(\text{Texas} \mid \text{Brand A}) \approx 0.64$[/tex].



Answer :

To determine if being from Texas and preferring Brand A are independent events, we need to compare the probability of being from Texas [tex]\( P(\text{Texas}) \)[/tex] with the conditional probability of being from Texas given that a person prefers Brand A [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex].

First, let's calculate [tex]\( P(\text{Texas}) \)[/tex]:

[tex]\[ P(\text{Texas}) = \frac{\text{Number of people from Texas}}{\text{Total number of people}} = \frac{125}{275} \approx 0.4545 \][/tex]

Next, let's calculate [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex]:

[tex]\[ P(\text{Texas} \mid \text{Brand A}) = \frac{\text{Number of Texans preferring Brand A}}{\text{Number of people preferring Brand A}} = \frac{80}{176} \approx 0.4545 \][/tex]

Now, let's compare the probabilities:
- [tex]\( P(\text{Texas}) \approx 0.4545 \)[/tex]
- [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.4545 \)[/tex]

If [tex]\( P(\text{Texas}) \)[/tex] is equal to [tex]\( P(\text{Texas} \mid \text{Brand A}) \)[/tex], it indicates that the events are independent. Since both probabilities are approximately equal, [tex]\( P(\text{Texas}) \approx P(\text{Texas} \mid \text{Brand A}) \)[/tex], we conclude that the events are independent.

Therefore, the answer is:

B. Yes, they are independent because [tex]\( P(\text{Texas}) \approx 0.45 \)[/tex] and [tex]\( P(\text{Texas} \mid \text{Brand A}) \approx 0.45 \)[/tex].