select the correct answer.

gavin collects a certain type of seashell. he has 160 seashells. the seashells weights are distributed normally, with a mean of 120 grams and a standard deviation of 26 grams.

how many of the seashells weigh at least 68 grams?

109
156
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134

Question

select the correct answer.

gavin collects a certain type of seashell. he has 160 seashells. the seashells weights are distributed normally, with a mean of 120 grams and a standard deviation of 26 grams.

how many of the seashells weigh at least 68 grams?

109
156
152
134

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Empirical Rule" ], "new_concepts": [ "Normal Distribution Calculations", "Z-Score Calculation" ], "current_concepts": [ "Empirical Rule", "Normal Distribution Calculations", "Z-Score Calculation" ] } ## Calculate the z-score for the threshold weight $$ \begin{aligned} &\mu = 120,\quad \sigma = 26,\quad X = 68\ &z = \frac{X - \mu}{\sigma} = \frac{68 - 120}{26} = \frac{-52}{26} = -2 \end{aligned} $$ ## Determine the proportion of seashells weighing at least 68 grams $$ \begin{aligned} &P(X \ge 68) = P(Z \ge -2)\ &P(-2 \le Z \le 2) = 0.95 \implies P(Z < -2) = \frac{1 - 0.95}{2} = 0.025\ &P(Z \ge -2) = 1 - 0.025 = 0.975 \end{aligned} $$ ## Calculate the expected number of seashells $$ \begin{aligned} &N = 160\ &\text{Count} = 160 \times 0.975 = 156 \end{aligned} $$

109

156

152

134

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Empirical Rule" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate the z-score for the threshold weight

Determine the proportion of seashells weighing at least 68 grams

Calculate the expected number of seashells

Answer: