a manufacturer knows that their items have a lengths that are skewed right, with a mean of 16.3 inches, and standard deviation of 3.1 inches.

if 34 items are chosen at random, what is the probability that their mean length is greater than 15.8 inches?

(round answer to four decimal places)

Question

a manufacturer knows that their items have a lengths that are skewed right, with a mean of 16.3 inches, and standard deviation of 3.1 inches.

if 34 items are chosen at random, what is the probability that their mean length is greater than 15.8 inches?

(round answer to four decimal places)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Central Limit Theorem", "Sampling Distribution", "Standard Error", "Normal Distribution Probability" ], "new_concepts": [], "current_concepts": [ "Central Limit Theorem", "Sampling Distribution", "Standard Error", "Normal Distribution Probability" ] } ## Identify the parameters Using the *Sampling Distribution* knowledge point $$ \begin{aligned} \mu &= 16.3 \ \sigma &= 3.1 \ n &= 34 \ \bar{X} &> 15.8 \end{aligned} $$ ## Apply Central Limit Theorem Using the *Central Limit Theorem* and *Standard Error* knowledge points $$ \begin{aligned} \mu_{\bar{x}} &= \mu = 16.3 \ \sigma_{\bar{x}} &= \frac{\sigma}{\sqrt{n}} = \frac{3.1}{\sqrt{34}} \approx 0.53164 \end{aligned} $$ ## Calculate the z-score Using the *Central Limit Theorem* knowledge point $$ \begin{aligned} z &= \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \ z &= \frac{15.8 - 16.3}{0.53164} \approx -0.9405 \end{aligned} $$ ## Find the probability Using the *Normal Distribution Probability* knowledge point $$ \begin{aligned} P(\bar{X} > 15.8) &= P(Z > -0.9405) \ &= 1 - P(Z \le -0.9405) \ &\approx 1 - 0.1735 \ &= 0.8265 \end{aligned} $$ A manufacturer knows that their items have a lengths that are skewed right, with a mean of 16.3 inches, and standard deviation of 3.1 inches. If 34 items are chosen at random, what is the probability that their mean length is greater than 15.8 inches? (Round answer to four decimal places) 0.8265 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Central Limit Theorem" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the parameters

Apply Central Limit Theorem

Calculate the z-score

Find the probability

Answer: