question 8 (multiple choice worth 10 points)
(02.04h mc)

a study shows that 70% of the population of a city have brown eyes and 30% do not. the study also shows that 10% of the people who do not have brown eyes do not have brown hair, while 80% of the people who have brown eyes have brown hair. a relative frequency table was constructed for this data. what is the largest marginal relative frequency and its interpretation?

0.56; individuals with brown eyes
0.70; individuals with brown eyes
0.80; individuals with brown hair
0.83; individuals with brown hair

Question

question 8 (multiple choice worth 10 points)
(02.04h mc)

a study shows that 70% of the population of a city have brown eyes and 30% do not. the study also shows that 10% of the people who do not have brown eyes do not have brown hair, while 80% of the people who have brown eyes have brown hair. a relative frequency table was constructed for this data. what is the largest marginal relative frequency and its interpretation?

0.56; individuals with brown eyes
0.70; individuals with brown eyes
0.80; individuals with brown hair
0.83; individuals with brown hair

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Marginal Relative Frequency", "Joint Relative Frequency" ], "current_concepts": [ "Two-Way Frequency Tables", "Marginal Relative Frequency", "Joint Relative Frequency" ] } ## Calculate joint relative frequencies $$ \begin{aligned} P(\text{Brown Eyes}) &= 0.70 \ P(\text{Not Brown Eyes}) &= 0.30 \ P(\text{Not Brown Hair} \mid \text{Not Brown Eyes}) &= 0.10 \implies P(\text{Not Brown Eyes} \cap \text{Not Brown Hair}) = 0.30 \times 0.10 = 0.03 \ P(\text{Brown Hair} \mid \text{Not Brown Eyes}) &= 0.90 \implies P(\text{Not Brown Eyes} \cap \text{Brown Hair}) = 0.30 \times 0.90 = 0.27 \ P(\text{Brown Hair} \mid \text{Brown Eyes}) &= 0.80 \implies P(\text{Brown Eyes} \cap \text{Brown Hair}) = 0.70 \times 0.80 = 0.56 \ P(\text{Not Brown Hair} \mid \text{Brown Eyes}) &= 0.20 \implies P(\text{Brown Eyes} \cap \text{Not Brown Hair}) = 0.70 \times 0.20 = 0.14 \end{aligned} $$ ## Calculate marginal relative frequencies $$ \begin{aligned} P(\text{Brown Eyes}) &= 0.70 \ P(\text{Not Brown Eyes}) &= 0.30 \ P(\text{Brown Hair}) &= P(\text{Brown Eyes} \cap \text{Brown Hair}) + P(\text{Not Brown Eyes} \cap \text{Brown Hair}) = 0.56 + 0.27 = 0.83 \ P(\text{Not Brown Hair}) &= P(\text{Brown Eyes} \cap \text{Not Brown Hair}) + P(\text{Not Brown Eyes} \cap \text{Not Brown Hair}) = 0.14 + 0.03 = 0.17 \end{aligned} $$ ## Identify the largest marginal relative frequency $$ \begin{aligned} \text{Marginal frequencies: } & \{0.70, 0.30, 0.83, 0.17\} \ \max(\text{Marginal frequencies}) &= 0.83 \quad (\text{Individuals with brown hair}) \end{aligned} $$

0.56; Individuals with brown eyes

0.70; Individuals with brown eyes

0.80; Individuals with brown hair

0.83; Individuals with brown hair

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Two-Way Frequency Tables" ] }

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Question

Explanation:

Calculate joint relative frequencies

Calculate marginal relative frequencies

Identify the largest marginal relative frequency

Answer: