Answer :
To determine the correct statement about the independence or dependence of events, we need to examine the relationships between the given preferences and age groups. We will start with the relevant probabilities:
1. Total Respondents: 183
2. Preferences and Age Groups:
[tex]\[ \begin{array}{|c|c|c|c|} \cline{2-4} \multicolumn{1}{c|}{} & \text{Age Below 20} & \text{Age 20 or Above} & \text{Total} \\ \hline \text{Text Articles} & 16 & 45 & 61 \\ \hline \text{Videos} & 32 & 90 & 122 \\ \hline \text{Total} & 48 & 135 & 183 \\ \hline \end{array} \][/tex]
### Let's compute the probabilities:
- Probability of preferring text articles ([tex]\(P(\text{Text})\)[/tex]):
[tex]\[ P(\text{Text}) = \frac{61}{183} \][/tex]
- Probability of being below 20 years old ([tex]\(P(\text{Below 20})\)[/tex]):
[tex]\[ P(\text{Below 20}) = \frac{48}{183} \][/tex]
- Probability of preferring text articles and being below 20 years old ([tex]\(P(\text{Text and Below 20})\)[/tex]):
[tex]\[ P(\text{Text and Below 20}) = \frac{16}{183} \][/tex]
### Check Independence:
Two events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if:
[tex]\[ P(A \, \text{and} \, B) = P(A) \times P(B) \][/tex]
1. For statement A (Independence of Text Articles and Below 20):
[tex]\[ P(\text{Text and Below 20}) = P(\text{Text}) \times P(\text{Below 20}) \][/tex]
Substitute the values:
[tex]\[ \frac{16}{183} \stackrel{?}{=} \left(\frac{61}{183}\right) \times \left(\frac{48}{183}\right) \][/tex]
Compute the product on the right side:
[tex]\[ \left(\frac{61}{183}\right) \times \left(\frac{48}{183}\right) = \frac{61 \times 48}{183 \times 183} \][/tex]
Simplify the fraction:
[tex]\[ \frac{2928}{33489} \approx \frac{16}{183} \][/tex]
Since:
[tex]\[ \frac{16}{183} \approx \frac{16}{183} \][/tex]
Both sides are equal, which indicates that preferring text articles and being below 20 are independent events.
Since this statement has been verified to be true, we can conclude:
### The correct statement is:
A. A respondent preferring text articles and a respondent being younger than 20 are independent events.
1. Total Respondents: 183
2. Preferences and Age Groups:
[tex]\[ \begin{array}{|c|c|c|c|} \cline{2-4} \multicolumn{1}{c|}{} & \text{Age Below 20} & \text{Age 20 or Above} & \text{Total} \\ \hline \text{Text Articles} & 16 & 45 & 61 \\ \hline \text{Videos} & 32 & 90 & 122 \\ \hline \text{Total} & 48 & 135 & 183 \\ \hline \end{array} \][/tex]
### Let's compute the probabilities:
- Probability of preferring text articles ([tex]\(P(\text{Text})\)[/tex]):
[tex]\[ P(\text{Text}) = \frac{61}{183} \][/tex]
- Probability of being below 20 years old ([tex]\(P(\text{Below 20})\)[/tex]):
[tex]\[ P(\text{Below 20}) = \frac{48}{183} \][/tex]
- Probability of preferring text articles and being below 20 years old ([tex]\(P(\text{Text and Below 20})\)[/tex]):
[tex]\[ P(\text{Text and Below 20}) = \frac{16}{183} \][/tex]
### Check Independence:
Two events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent if:
[tex]\[ P(A \, \text{and} \, B) = P(A) \times P(B) \][/tex]
1. For statement A (Independence of Text Articles and Below 20):
[tex]\[ P(\text{Text and Below 20}) = P(\text{Text}) \times P(\text{Below 20}) \][/tex]
Substitute the values:
[tex]\[ \frac{16}{183} \stackrel{?}{=} \left(\frac{61}{183}\right) \times \left(\frac{48}{183}\right) \][/tex]
Compute the product on the right side:
[tex]\[ \left(\frac{61}{183}\right) \times \left(\frac{48}{183}\right) = \frac{61 \times 48}{183 \times 183} \][/tex]
Simplify the fraction:
[tex]\[ \frac{2928}{33489} \approx \frac{16}{183} \][/tex]
Since:
[tex]\[ \frac{16}{183} \approx \frac{16}{183} \][/tex]
Both sides are equal, which indicates that preferring text articles and being below 20 are independent events.
Since this statement has been verified to be true, we can conclude:
### The correct statement is:
A. A respondent preferring text articles and a respondent being younger than 20 are independent events.
