A simple random sample of size n = 49 is obtained from a population with µ= 80 and σ = 14. NOTE: Round your answer to four decimal places. (a) What is P > > 83 )? (b) What is P ≤ 75.8)? (c) What is P(78.3< < 85.1)?
A simple random sample of size n = 49 is obtained from a population with µ= 80 and σ = 14. NOTE: Round your answer to four decimal places. (a) What is P > > 83 )? (b) What is P ≤ 75.8)? (c) What is P(78.3< < 85.1)?
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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![### Sample Size Probability Calculations
A simple random sample of size \( n = 49 \) is obtained from a population with a mean \( \mu = 80 \) and a standard deviation \( \sigma = 14 \).
**NOTE:** Round your answer to four decimal places.
#### Questions:
1. **(a)** What is \( P\left(\bar{X} > 83\right) \)? \_\_\_\_\_
2. **(b)** What is \( P\left(\bar{X} \leq 75.8\right) \)? \_\_\_\_\_
3. **(c)** What is \( P\left(78.3 < \bar{X} < 85.1\right) \)? \_\_\_\_\_
---
In solving these probability questions, you will need to use the properties of the sampling distribution of the sample mean. Recall that if the population is normally distributed, or if the sample size is sufficiently large, the distribution of the sample mean will approximately follow a normal distribution with:
- **Mean of the sampling distribution ( \( \mu_{\bar{X}} \) ):** Equal to the population mean (\( \mu \)).
- **Standard error of the mean ( \( \sigma_{\bar{X}} \) ):** Equal to the population standard deviation (\( \sigma \)) divided by the square root of the sample size (\( n \)):
\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2297071-c636-4616-baf3-32dd51d4726a%2F28ca18a7-47cf-4818-b95b-dc30f62e62d1%2Fezwotfb_processed.png&w=3840&q=75)
Transcribed Image Text:### Sample Size Probability Calculations
A simple random sample of size \( n = 49 \) is obtained from a population with a mean \( \mu = 80 \) and a standard deviation \( \sigma = 14 \).
**NOTE:** Round your answer to four decimal places.
#### Questions:
1. **(a)** What is \( P\left(\bar{X} > 83\right) \)? \_\_\_\_\_
2. **(b)** What is \( P\left(\bar{X} \leq 75.8\right) \)? \_\_\_\_\_
3. **(c)** What is \( P\left(78.3 < \bar{X} < 85.1\right) \)? \_\_\_\_\_
---
In solving these probability questions, you will need to use the properties of the sampling distribution of the sample mean. Recall that if the population is normally distributed, or if the sample size is sufficiently large, the distribution of the sample mean will approximately follow a normal distribution with:
- **Mean of the sampling distribution ( \( \mu_{\bar{X}} \) ):** Equal to the population mean (\( \mu \)).
- **Standard error of the mean ( \( \sigma_{\bar{X}} \) ):** Equal to the population standard deviation (\( \sigma \)) divided by the square root of the sample size (\( n \)):
\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]
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