Alex wants to find out: if the standard deviation of the mean for the sampling distribution of random samples of size 256 from a large or infinite population is 9, how large must the sample size become if the standard deviation is to be reduced to 2.7 (or even smaller)?
Alex wants to find out: if the standard deviation of the mean for the sampling distribution of random samples of size 256 from a large or infinite population is 9, how large must the sample size become if the standard deviation is to be reduced to 2.7 (or even smaller)?
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 22PFA
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Question
![**Problem Statement:**
Alex wants to find out: if the standard deviation of the mean for the sampling distribution of random samples of size 256 from a large or infinite population is 9, how large must the sample size become if the standard deviation is to be reduced to 2.7 (or even smaller)?
**Solution Explanation:**
In statistics, the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is calculated using the formula:
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]
Where:
- \(\sigma_{\bar{x}}\) is the standard deviation of the sampling distribution (standard error).
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
Given that \(\sigma_{\bar{x}} = 9\) for \(n = 256\), we can find \(\sigma\):
\[
9 = \frac{\sigma}{\sqrt{256}} \Rightarrow \sigma = 9 \times 16 = 144
\]
Now, we need to find the sample size \(n\) such that \(\sigma_{\bar{x}}\) is reduced to 2.7:
\[
2.7 = \frac{144}{\sqrt{n}}
\]
Solve for \(n\):
\[
2.7\sqrt{n} = 144 \Rightarrow \sqrt{n} = \frac{144}{2.7}
\]
Square both sides to solve for \(n\):
\[
n = \left(\frac{144}{2.7}\right)^2 = \left(\frac{144}{2.7}\right)^2 \approx 2844.44
\]
Therefore, the sample size must be approximately 2845 to reduce the standard deviation to 2.7 or smaller.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d14b178-5b8c-4639-9880-9a6310b5f0da%2Fbf00c220-5926-42ba-9969-fb79c78fd144%2F97qv16j_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Alex wants to find out: if the standard deviation of the mean for the sampling distribution of random samples of size 256 from a large or infinite population is 9, how large must the sample size become if the standard deviation is to be reduced to 2.7 (or even smaller)?
**Solution Explanation:**
In statistics, the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is calculated using the formula:
\[
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
\]
Where:
- \(\sigma_{\bar{x}}\) is the standard deviation of the sampling distribution (standard error).
- \(\sigma\) is the population standard deviation.
- \(n\) is the sample size.
Given that \(\sigma_{\bar{x}} = 9\) for \(n = 256\), we can find \(\sigma\):
\[
9 = \frac{\sigma}{\sqrt{256}} \Rightarrow \sigma = 9 \times 16 = 144
\]
Now, we need to find the sample size \(n\) such that \(\sigma_{\bar{x}}\) is reduced to 2.7:
\[
2.7 = \frac{144}{\sqrt{n}}
\]
Solve for \(n\):
\[
2.7\sqrt{n} = 144 \Rightarrow \sqrt{n} = \frac{144}{2.7}
\]
Square both sides to solve for \(n\):
\[
n = \left(\frac{144}{2.7}\right)^2 = \left(\frac{144}{2.7}\right)^2 \approx 2844.44
\]
Therefore, the sample size must be approximately 2845 to reduce the standard deviation to 2.7 or smaller.
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