A random variable is normally distributed. It has a mean of 201 and a standard deviation of 15. If you take a sample of size 15, can you say what the shape of the sampling distribution for the sample mean is? Why? OT If the sample size is 15, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed. If the sample size is 15, then you can say the sampling distribution of the sample mean is not normally distributed since the sample size is less than 30. O If the sample size is 15, then you can't say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is less than 30. 0x = For a sample of size 15, state the mean of the sample mean and the standard deviation of the sample mean. Round to two decimal places if necessary. Hx = 201 Or For a sample of size 15, find the probability that the sample mean is more than 200. P(x > 200) =
A random variable is normally distributed. It has a mean of 201 and a standard deviation of 15. If you take a sample of size 15, can you say what the shape of the sampling distribution for the sample mean is? Why? OT If the sample size is 15, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed. If the sample size is 15, then you can say the sampling distribution of the sample mean is not normally distributed since the sample size is less than 30. O If the sample size is 15, then you can't say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is less than 30. 0x = For a sample of size 15, state the mean of the sample mean and the standard deviation of the sample mean. Round to two decimal places if necessary. Hx = 201 Or For a sample of size 15, find the probability that the sample mean is more than 200. P(x > 200) =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![**Understanding Sampling Distribution and Calculations**
A random variable is normally distributed with a mean of 201 and a standard deviation of 15.
**Question:**
If you take a sample of size 15, can you say what the shape of the sampling distribution for the sample mean is? Why?
**Answer Options:**
1. **Correct:** If the sample size is 15, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed.
2. If the sample size is 15, then you can say the sampling distribution of the sample mean is not normally distributed since the sample size is less than 30.
3. If the sample size is 15, then you can't say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is less than 30.
**Explanation:**
The correct choice is highlighted to indicate that with a normally distributed variable, the sampling distribution of the sample mean will also be normally distributed, even with a smaller sample size.
---
**Calculations:**
For a sample size of 15, state the mean of the sample mean and the standard deviation of the sample mean. Round to two decimal places if necessary.
- \(\mu_{\bar{x}}\) = 201 ✓
- \(\sigma_{\bar{x}} =\)
**Probability Question:**
For a sample of size 15, find the probability that the sample mean is more than 200.
- \(P(\bar{x} > 200) = \)
**Notes:**
- \(\mu_{\bar{x}}\) (mean of the sample mean) is equal to the population mean, 201.
- \(\sigma_{\bar{x}}\) (standard deviation of the sample mean) needs to be calculated using the formula:
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]
where \(\sigma\) is the standard deviation of the population (15) and \(n\) is the sample size (15).
The probability calculation requires further work using statistical methods such as Z-scores.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88c4a12b-1ea9-44c1-9400-53c8c9c9bde6%2F853df59e-da19-41ee-84ab-97cedc923096%2Fh6etkw_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Sampling Distribution and Calculations**
A random variable is normally distributed with a mean of 201 and a standard deviation of 15.
**Question:**
If you take a sample of size 15, can you say what the shape of the sampling distribution for the sample mean is? Why?
**Answer Options:**
1. **Correct:** If the sample size is 15, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed.
2. If the sample size is 15, then you can say the sampling distribution of the sample mean is not normally distributed since the sample size is less than 30.
3. If the sample size is 15, then you can't say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is less than 30.
**Explanation:**
The correct choice is highlighted to indicate that with a normally distributed variable, the sampling distribution of the sample mean will also be normally distributed, even with a smaller sample size.
---
**Calculations:**
For a sample size of 15, state the mean of the sample mean and the standard deviation of the sample mean. Round to two decimal places if necessary.
- \(\mu_{\bar{x}}\) = 201 ✓
- \(\sigma_{\bar{x}} =\)
**Probability Question:**
For a sample of size 15, find the probability that the sample mean is more than 200.
- \(P(\bar{x} > 200) = \)
**Notes:**
- \(\mu_{\bar{x}}\) (mean of the sample mean) is equal to the population mean, 201.
- \(\sigma_{\bar{x}}\) (standard deviation of the sample mean) needs to be calculated using the formula:
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]
where \(\sigma\) is the standard deviation of the population (15) and \(n\) is the sample size (15).
The probability calculation requires further work using statistical methods such as Z-scores.
Transcribed Image Text:### Educational Resource on Sample Mean Distribution
**Question:**
If you take a sample of size 37, can you say what the shape of the distribution of the sample mean is? Why?
**Options:**
- **Option 1 (Selected):** If the sample size is 37, then you can say the sampling distribution of the sample mean is normally distributed since the variable is normally distributed.
- **Option 2:** If the sample size is 37, then you can say the sampling distribution of the sample mean is not normally distributed since the sample size is greater than 30.
- **Option 3:** If the sample size is 37, then you can't say anything about the sampling distribution of the sample mean, since the population of the random variable is not normally distributed and the sample size is greater than 30.
**Explanation:**
The correct option is highlighted. When the variable is normally distributed, the sampling distribution of the sample mean is normally distributed for any sample size, including 37.
---
**Task:**
For a sample of size 37, state the mean of the sample mean and the standard deviation of the sample mean. Round to two decimal places if necessary.
**Mean of Sample Mean (\( \mu_{\bar{x}} \)):**
\( \mu_{\bar{x}} = 201 \) (This value is provided and confirmed.)
**Standard Deviation of Sample Mean (\( \sigma_{\bar{x}} \)):**
A space is provided for the user to input the standard deviation.
---
**Probability Task:**
For a sample of size 37, find the probability that the sample mean is more than 200, denoted by \( P(\bar{x} > 200) \).
There is a blank space for users to calculate and provide the probability value.
---
### Summary
The above exercise helps in understanding the Central Limit Theorem, which suggests that as sample size increases, the sampling distribution of the sample mean approaches a normal distribution. This particular example assumes a normally distributed variable, making the sample mean's distribution normally distributed for any sample size, including 37. Calculations and probability assessments are based on this understanding.
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