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A simple random sample of size n = 36 is obtained from a population with μ = 72 and a = 18. (a) Describe the sampling distribution of x. (b) What is P (x>77.4)? (c) What is P (x≤64.65) ? (d) What is P (68.4 77.4)= (Round to four decimal places as needed.) www

A simple random sample of size n = 36 is obtained from a population with μ = 72 and a = 18. (a) Describe the sampling distribution of x. (b) What is P (x>77.4)? (c) What is P (x≤64.65) ? (d) What is P (68.4 77.4)= (Round to four decimal places as needed.) www

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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**Educational Website Content** ## Understanding Sampling Distribution A simple random sample of size \( n = 36 \) is obtained from a population with \( \mu = 72 \) and \( \sigma = 18 \). ### Questions and Solutions: --- #### (a) Describe the sampling distribution of \( \bar{x} \). - **Answer Choices:** - \( \quad \) B. The distribution is skewed right. - \( \quad \) C. The distribution is uniform. - \( \checkmark \) D. The distribution is approximately normal. - \( \quad \) E. The shape of the distribution is unknown. Explanation: When the sample size is sufficiently large, the sampling distribution of the sample mean \( \bar{x} \) is approximately normal due to the Central Limit Theorem. Hence, the correct answer is D. --- #### Find the mean and standard deviation of the sampling distribution of \( \bar{x} \). \[ \mu_{\bar{x}} = 72 \] \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{18}{\sqrt{36}} = 3 \] --- #### (b) What is \( P(\bar{x} > 77.4) \)? Calculate the probability, rounding to four decimal places as needed. \[\text{P}(\bar{x} > 77.4) = \] *To be calculated.* --- #### (c) What is \( P(\bar{x} \leq 64.65) \)? *This would also need to be calculated using the same methods.* --- #### (d) What is \( P(68.4 < \bar{x} < 77.7) \)? *This probability calculation would require the use of Z-scores and referencing standard normal distribution tables or a computational tool.* --- ### Explanation of Graphs and Diagrams: *There are no graphs or diagrams present in the image provided. The content primarily consists of textual information and calculations related to the sampling distribution of the sample mean.* --- *Note: For parts (b), (c), and (d), to find the specific probabilities, one would typically convert the sample mean values into Z-scores using the formula:* \[ Z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \
Transcribed Image Text:**Educational Website Content** ## Understanding Sampling Distribution A simple random sample of size \( n = 36 \) is obtained from a population with \( \mu = 72 \) and \( \sigma = 18 \). ### Questions and Solutions: --- #### (a) Describe the sampling distribution of \( \bar{x} \). - **Answer Choices:** - \( \quad \) B. The distribution is skewed right. - \( \quad \) C. The distribution is uniform. - \( \checkmark \) D. The distribution is approximately normal. - \( \quad \) E. The shape of the distribution is unknown. Explanation: When the sample size is sufficiently large, the sampling distribution of the sample mean \( \bar{x} \) is approximately normal due to the Central Limit Theorem. Hence, the correct answer is D. --- #### Find the mean and standard deviation of the sampling distribution of \( \bar{x} \). \[ \mu_{\bar{x}} = 72 \] \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{18}{\sqrt{36}} = 3 \] --- #### (b) What is \( P(\bar{x} > 77.4) \)? Calculate the probability, rounding to four decimal places as needed. \[\text{P}(\bar{x} > 77.4) = \] *To be calculated.* --- #### (c) What is \( P(\bar{x} \leq 64.65) \)? *This would also need to be calculated using the same methods.* --- #### (d) What is \( P(68.4 < \bar{x} < 77.7) \)? *This probability calculation would require the use of Z-scores and referencing standard normal distribution tables or a computational tool.* --- ### Explanation of Graphs and Diagrams: *There are no graphs or diagrams present in the image provided. The content primarily consists of textual information and calculations related to the sampling distribution of the sample mean.* --- *Note: For parts (b), (c), and (d), to find the specific probabilities, one would typically convert the sample mean values into Z-scores using the formula:* \[ Z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \
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