consider a population with $\mu = 137$ and $\sigma = 27$ and a sampling distribution with sample size 70. (a) what are the mean and standard deviation of the sampling distribution? round standard deviation to four decimal places. mean: and standard deviation: (b) use the normal distribution tool to determine the following probabilities: enter the probability as a percentage rounded to 2 decimal places. do not enter the $\%$ symbol. for example, if probability is 0.12345 you would enter the response 12.35. $p(\bar{x} 143) =$ $\%$ $p(\bar{x}

Question

consider a population with $\mu = 137$ and $\sigma = 27$ and a sampling distribution with sample size 70.

(a) what are the mean and standard deviation of the sampling distribution?

round standard deviation to four decimal places.

mean:   and standard deviation:

(b) use the normal distribution tool to determine the following probabilities:

enter the probability as a percentage rounded to 2 decimal places. do not enter the $\%$ symbol. for example, if probability is 0.12345 you would enter the response 12.35.

$p(\bar{x} > 143) =$   $\%$

$p(\bar{x} < 133) =$   $\%$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Sampling Distribution", "Standard Error" ], "new_concepts": [ "Normal Distribution Probability" ], "current_concepts": [ "Sampling Distribution", "Standard Error", "Central Limit Theorem", "Normal Distribution Probability" ] } ## Calculate mean and standard error Using the *Sampling Distribution* and *Standard Error* knowledge points $$ \begin{aligned} \mu_{\bar{x}} &= \mu = 137 \ \sigma_{\bar{x}} &= \frac{\sigma}{\sqrt{n}} = \frac{27}{\sqrt{70}} \approx 3.2271 \end{aligned} $$ ## Calculate probability for x-bar greater than 143 Find the z-score and the corresponding upper-tail probability. $$ \begin{aligned} z_1 &= \frac{143 - 137}{3.2271} \approx 1.8593 \ P(\bar{x} > 143) &= P(Z > 1.8593) \approx 0.0315 \approx 3.15\% \end{aligned} $$ ## Calculate probability for x-bar less than 133 Find the z-score and the corresponding lower-tail probability. $$ \begin{aligned} z_2 &= \frac{133 - 137}{3.2271} \approx -1.2395 \ P(\bar{x} < 133) &= P(Z < -1.2395) \approx 0.1076 \approx 10.76\% \end{aligned} $$ ## Question 1 (a) What are the mean and standard deviation of the sampling distribution? Mean: 137 and Standard Deviation: 3.2271 ## Question 2 (b) Use the Normal Distribution tool to determine the following probabilities: $P(\bar{x} > 143) =$ 3.15 % $P(\bar{x} < 133) =$ 10.76 % { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Statistics", "Sampling Distribution" ] }

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Explanation:

Calculate mean and standard error

Calculate probability for x-bar greater than 143

Calculate probability for x-bar less than 133

Answer:

Question 1

Question 2