the x distribution to have a sample mean greater than 9? Explain. Basic Computation: Central Limit Theorem Suppose x has a distribution with a mean of 20 and a standard deviation of 3. Random samples of size n = 36 are drawn. (a) Describe the x distribution and compute the mean and standard deviation of the distribution. (b) Find the z value corresponding to x = 19. (c) Find P(x< 19). (d) Interpretation Would it be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 19? Explain.
the x distribution to have a sample mean greater than 9? Explain. Basic Computation: Central Limit Theorem Suppose x has a distribution with a mean of 20 and a standard deviation of 3. Random samples of size n = 36 are drawn. (a) Describe the x distribution and compute the mean and standard deviation of the distribution. (b) Find the z value corresponding to x = 19. (c) Find P(x< 19). (d) Interpretation Would it be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 19? Explain.
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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Answer number 6. All steps/parts! Thank you!
Transcribed Image Text:### Basic Computation: Central Limit Theorem
#### Problem 5
Suppose \( x \) has a distribution with a mean of 8 and a standard deviation of 16. Random samples of size \( n = 64 \) are drawn.
**(a)** Describe the \( \bar{x} \) distribution and compute the mean and standard deviation of the distribution.
**(b)** Find the \( z \) value corresponding to \( \bar{x} = 9 \).
**(c)** Find \( P(\bar{x} > 9) \).
**(d)** **Interpretation:** Would it be unusual for a random sample of size 64 from the \( x \) distribution to have a sample mean greater than 9? Explain.
#### Problem 6
Suppose \( x \) has a distribution with a mean of 20 and a standard deviation of 3. Random samples of size \( n = 36 \) are drawn.
**(a)** Describe the \( \bar{x} \) distribution and compute the mean and standard deviation of the distribution.
**(b)** Find the \( z \) value corresponding to \( \bar{x} = 19 \).
**(c)** Find \( P(\bar{x} < 19) \).
**(d)** **Interpretation:** Would it be unusual for a random sample of size 36 from the \( x \) distribution to have a sample mean less than 19? Explain.
### Detailed Explanation
1. **Distribution and Computation**:
- The distribution of the sample mean \( \bar{x} \) approaches a normal distribution as the sample size increases, due to the Central Limit Theorem.
- To compute the mean and standard deviation of the \( \bar{x} \) distribution:
- The mean of the sampling distribution (\( \mu_{\bar{x}} \)) is equal to the mean of the population (\( \mu \)).
- The standard deviation of the sampling distribution (\( \sigma_{\bar{x}} \)) is equal to the population standard deviation (\( \sigma \)) divided by the square root of the sample size (\( n \)).
2. **Z Value Calculation**:
- The \( z \) value for a given \( \bar{x} \) can be found using the formula:
\[
z = \frac{\bar{x} -
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