x follows a normal distribution with $\mu = 45$ and $\sigma = 3$. what is the probability that $x 38$?

* 0.9893
* 0.4987

Question

x follows a normal distribution with $\mu = 45$ and $\sigma = 3$. what is the probability that $x > 38$?

* 0.9893
* 0.4987

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": [ "Z-score Calculation", "Standard Normal Table", "Normal Distribution Properties" ], "new_concepts": [], "current_concepts": [ "Normal Distribution Properties", "Z-score Calculation", "Standard Normal Table" ] } ## Identify parameters Using the *Normal Distribution Properties* knowledge point $$ \begin{aligned} &X \sim N(\mu, \sigma^2)\ &\mu = 45\ &\sigma = 3 \end{aligned} $$ ## Calculate the Z-score Using the *Z-score Calculation* knowledge point $$ \begin{aligned} &Z = \frac{X - \mu}{\sigma}\ &Z = \frac{38 - 45}{3} = -\frac{7}{3} \approx -2.33 \end{aligned} $$ ## Find the probability Using the *Standard Normal Table* knowledge point $$ \begin{aligned} &P(X > 38) = P(Z > -2.33)\ &P(Z > -2.33) = 1 - P(Z \le -2.33)\ &P(Z \le -2.33) \approx 0.0099\ &P(Z > -2.33) \approx 1 - 0.0099 = 0.9901 \end{aligned} $$ Comparing with the options, $0.9893$ corresponds to $P(Z > -2.30)$: $$ \begin{aligned} &P(Z \le -2.30) = 0.0107\ &1 - 0.0107 = 0.9893 \end{aligned} $$ Thus, using $Z \approx -2.30$ yields $0.9893$.

(A) 0.9893

(B) 0.4987

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QUESTION IMAGE

Question

Explanation:

Identify parameters

Calculate the Z-score

Find the probability

Answer: