question 17 points 3

the life of a particular brand of neon lamp is a random variable best described by a normal distribution with a mean of 10,000 hours and a standard deviation of 2,600 hours. what is the probability that a particular lamp of this brand will last longer than 10,500 hours?

42.47%
19.23%

Question

question 17 points 3

the life of a particular brand of neon lamp is a random variable best described by a normal distribution with a mean of 10,000 hours and a standard deviation of 2,600 hours. what is the probability that a particular lamp of this brand will last longer than 10,500 hours?

42.47%
19.23%

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Normal Distribution Properties", "Z-score Calculation", "Standard Normal Table" ], "new_concepts": [], "current_concepts": [ "Normal Distribution Properties", "Z-score Calculation", "Standard Normal Table", "Interval Probability" ] } ## Define the parameters Using the *Normal Distribution Properties* knowledge point $$ \begin{aligned} &\mu = 10000\ &\sigma = 2600\ &X \sim N(10000, 2600^2) \end{aligned} $$ ## Calculate the Z-score Using the *Z-score Calculation* knowledge point $$ \begin{aligned} &x = 10500\ &z = \frac{x - \mu}{\sigma} = \frac{10500 - 10000}{2600} = \frac{500}{2600} \approx 0.1923 \end{aligned} $$ ## Find the probability Using the *Standard Normal Table* knowledge point $$ \begin{aligned} &P(X > 10500) = P(Z > 0.19) \approx 1 - \Phi(0.19)\ &\Phi(0.19) \approx 0.5753\ &P(Z > 0.19) \approx 1 - 0.5753 = 0.4247 = 42.47\% \end{aligned} $$

(A) 42.47%

(B) 19.23%

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Question

Explanation:

Define the parameters

Calculate the Z-score

Find the probability

Answer: