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Without doing any computation, put the following in order from least to greatest, assuming the population is normally distributed with = 100 and a = 15. (a) P(90 sxs 110) for a random sample of size n = 50 (b) P(90 sxs 110) for a random sample of size n = 30 (c) P(90≤x≤ 110) <

Without doing any computation, put the following in order from least to greatest, assuming the population is normally distributed with = 100 and a = 15. (a) P(90 sxs 110) for a random sample of size n = 50 (b) P(90 sxs 110) for a random sample of size n = 30 (c) P(90≤x≤ 110) <

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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### Example Problem on Normal Distribution and Probability: Without doing any computation, put the following in order from least to greatest, assuming the population is normally distributed with μ = 100 and σ = 15: (a) P(90 ≤ x ≤ 110) for a random sample of size n = 50 (b) P(90 ≤ x ≤ 110) for a random sample of size n = 10 (c) P(90 ≤ x ≤ 110) for a random sample of size n = 30 ### Explanation: In this problem, we are asked to compare the probabilities of x falling between 90 and 110 for different sample sizes from a normally distributed population. The mean (μ) of the population is 100, and the standard deviation (σ) is 15. When we take samples from a population, the distribution of the sample mean will be normally distributed due to the Central Limit Theorem (CLT). The standard error of the mean (σ_x̄) equals the population standard deviation divided by the square root of the sample size (σ/√n). As the sample size increases, the standard error decreases, making the sample mean more concentrated around the population mean. This means that the range (90 to 110) will likely include a higher probability with larger sample sizes. Therefore, you should expect the probabilities to be ordered from least to greatest as follows: (b) P(90 ≤ x ≤ 110) for n = 10 (c) P(90 ≤ x ≤ 110) for n = 30 (a) P(90 ≤ x ≤ 110) for n = 50
Transcribed Image Text:### Example Problem on Normal Distribution and Probability: Without doing any computation, put the following in order from least to greatest, assuming the population is normally distributed with μ = 100 and σ = 15: (a) P(90 ≤ x ≤ 110) for a random sample of size n = 50 (b) P(90 ≤ x ≤ 110) for a random sample of size n = 10 (c) P(90 ≤ x ≤ 110) for a random sample of size n = 30 ### Explanation: In this problem, we are asked to compare the probabilities of x falling between 90 and 110 for different sample sizes from a normally distributed population. The mean (μ) of the population is 100, and the standard deviation (σ) is 15. When we take samples from a population, the distribution of the sample mean will be normally distributed due to the Central Limit Theorem (CLT). The standard error of the mean (σ_x̄) equals the population standard deviation divided by the square root of the sample size (σ/√n). As the sample size increases, the standard error decreases, making the sample mean more concentrated around the population mean. This means that the range (90 to 110) will likely include a higher probability with larger sample sizes. Therefore, you should expect the probabilities to be ordered from least to greatest as follows: (b) P(90 ≤ x ≤ 110) for n = 10 (c) P(90 ≤ x ≤ 110) for n = 30 (a) P(90 ≤ x ≤ 110) for n = 50
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