What formula is used to gain information about a sample mean when the variable is normally distributed or when the sample size is 30 or more?
What formula is used to gain information about a sample mean when the variable is normally distributed or when the sample size is 30 or more?
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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![**7. What formula is used to gain information about a sample mean when the variable is normally distributed or when the sample size is 30 or more?**
This question pertains to statistical methods used to analyze sample data. It refers to the use of the Central Limit Theorem, which states that when the sample size is large (typically n ≥ 30), the sampling distribution of the sample mean can be approximated by a normal distribution, regardless of the shape of the population distribution.
When the variable is normally distributed or the sample size is 30 or more, the formula used to calculate the confidence interval for the sample mean is typically the Z-score formula for the mean:
\[
\bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)
\]
- \(\bar{x}\) is the sample mean.
- \(Z\) is the Z-value from the Z-table corresponding to the desired confidence level.
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
In cases where the population standard deviation is unknown, the sample standard deviation (s) is used, and the t-distribution is applied instead of the Z-distribution:
\[
\bar{x} \pm t \left(\frac{s}{\sqrt{n}}\right)
\]
Understanding these concepts allows researchers to make inferences about population parameters based on sample statistics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb254707-7c83-4423-a4c4-c458206b4db5%2F43c6b9da-df46-4114-926e-a2a9347a5be6%2Fjlthbw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**7. What formula is used to gain information about a sample mean when the variable is normally distributed or when the sample size is 30 or more?**
This question pertains to statistical methods used to analyze sample data. It refers to the use of the Central Limit Theorem, which states that when the sample size is large (typically n ≥ 30), the sampling distribution of the sample mean can be approximated by a normal distribution, regardless of the shape of the population distribution.
When the variable is normally distributed or the sample size is 30 or more, the formula used to calculate the confidence interval for the sample mean is typically the Z-score formula for the mean:
\[
\bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)
\]
- \(\bar{x}\) is the sample mean.
- \(Z\) is the Z-value from the Z-table corresponding to the desired confidence level.
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
In cases where the population standard deviation is unknown, the sample standard deviation (s) is used, and the t-distribution is applied instead of the Z-distribution:
\[
\bar{x} \pm t \left(\frac{s}{\sqrt{n}}\right)
\]
Understanding these concepts allows researchers to make inferences about population parameters based on sample statistics.
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