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Drivers were surveyed about their driving styles and the type of vehicle they drive. The results are summarized in the two-way table below:

\begin{tabular}{|l|l|l|l|}
\hline
& Attentive & Defensive & Eco-friendly \\
\hline
Sedan & 24 & 20 & 10 \\
\hline
SUV & 18 & 13 & 7 \\
\hline
Motorcycle & 12 & 5 & 7 \\
\hline
Truck & 6 & 10 & 4 \\
\hline
\end{tabular}

The equation [tex]$0.5 \neq 0.353$[/tex] is the mathematical justification proving which of the following statements?

A. The events driving in a defensive style and driving a truck are approximately independent because [tex]$P($[/tex]truck [tex]$|$[/tex] defensive [tex]$) \neq P($[/tex]truck[tex]$)$[/tex].

B. The events driving in a defensive style and driving a truck are not independent because [tex]$P($[/tex]defensive truck[tex]$) \neq P($[/tex]defensive[tex]$)$[/tex].

C. The events driving in a defensive style and driving a truck are approximately independent because [tex]$P($[/tex]defensive [tex]$|$[/tex] truck[tex]$) \neq P($[/tex]defensive[tex]$)$[/tex].

D. The events driving in a defensive style and driving a truck are not independent because [tex]$P($[/tex]truck defensive[tex]$) \neq P($[/tex]truck[tex]$)$[/tex].

Sagot :

GinnyAnswer
Let's work through the problem step by step and justify the correct statement based on the given results and probabilities.

### Step 1: Calculate the total number of drivers
Sum all the numbers in the table:
- Sedan: 24 (Attentive) + 20 (Defensive) + 10 (Eco-friendly) = 54
- SUV: 18 (Attentive) + 13 (Defensive) + 7 (Eco-friendly) = 38
- Motorcycle: 12 (Attentive) + 5 (Defensive) + 7 (Eco-friendly) = 24
- Truck: 6 (Attentive) + 10 (Defensive) + 4 (Eco-friendly) = 20

Total number of drivers = 54 + 38 + 24 + 20 = 136.

### Step 2: Calculate the total number of drivers for each style
- Defensive: 20 (Sedan) + 13 (SUV) + 5 (Motorcycle) + 10 (Truck) = 48

### Step 3: Calculate the total number of truck drivers
- Trucks: 6 (Attentive) + 10 (Defensive) + 4 (Eco-friendly) = 20

### Step 4: Calculate the probabilities
- Probability of driving a truck (P(Truck)):
[tex]\[ P(\text{Truck}) = \frac{20}{136} = 0.14705882352941177 \][/tex]

- Probability of driving in a defensive style (P(Defensive)):
[tex]\[ P(\text{Defensive}) = \frac{48}{136} = 0.35294117647058826 \][/tex]

- Probability of driving a truck given defensive style (P(Truck|Defensive)):
Total number of drivers driving trucks in a defensive style is 10.
[tex]\[ P(\text{Truck|Defensive}) = \frac{10}{48} = 0.20833333333333334 \][/tex]

### Step 5: Determine the relationship between these probabilities
According to the results:
- [tex]\( P(\text{Truck}) = 0.14705882352941177 \)[/tex]
- [tex]\( P(\text{Truck|Defensive}) = 0.20833333333333334 \)[/tex]

Since [tex]\( P(\text{Truck}) \neq P(\text{Truck|Defensive}) \)[/tex], the events of driving in a defensive style and driving a truck are not independent.

### Step 6: Validate which statement is correct

Comparing the results against the statements:

- The events driving in a defensive style and driving a truck are not independent because [tex]\( P(\text{Truck|Defensive}) \neq P(\text{Truck}) \)[/tex].

Therefore, the correct statement based on our calculations is:

- The events driving in a defensive style and driving a truck are not independent because [tex]\( P(\text{Truck|Defensive}) \neq P(\text{Truck}) \)[/tex].
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