Textbook Question
"Answer the following questions for the sampling distribution of the sample mean shown to the right.
c. If the sample size is n = 16, what is the standard deviation of the population from which the sample was drawn?"
42
views
7. Sampling Distributions & Confidence Intervals: Mean
Sampling Distribution of the Sample Mean and Central Limit Theorem
3:19 minutesProblem 8.1.6
Textbook Question
True or False: To cut the standard error of the mean in half, the sample size must be doubled.
Verified step by step guidance1
Recall the formula for the standard error of the mean (SEM), which is given by:
\[\text{SEM} = \frac{\sigma}{\sqrt{n}}\]
where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
Understand that the SEM is inversely proportional to the square root of the sample size, meaning that if you want to change the SEM, you need to adjust \(n\) accordingly.
Set up the relationship to find the new sample size \(n_{new}\) that will cut the SEM in half:
\[\frac{\sigma}{\sqrt{n_{new}}} = \frac{1}{2} \times \frac{\sigma}{\sqrt{n}}\]
Simplify the equation by canceling \(\sigma\) and rearranging terms:
\[\frac{1}{\sqrt{n_{new}}} = \frac{1}{2} \times \frac{1}{\sqrt{n}} \implies \sqrt{n_{new}} = 2 \sqrt{n}\]
Square both sides to solve for \(n_{new}\):
\[n_{new} = (2 \sqrt{n})^2 = 4n\]
This shows that to cut the SEM in half, the sample size must be quadrupled, not just doubled.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Error of the Mean (SEM)
The standard error of the mean measures the variability of sample means around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size, reflecting how sample size affects estimate precision.
Recommended video:
Guided course
04:52
Calculating the Mean
Relationship Between Sample Size and SEM
The SEM decreases as the sample size increases, specifically by the square root of the sample size. Doubling the sample size reduces the SEM by a factor of 1 divided by the square root of 2, not by half.
Recommended video:
04:08Finding the Minimum Sample Size Needed for a Confidence Interval
Implications for Sample Size Planning
To reduce the SEM by half, the sample size must be increased by a factor of four, since SEM is inversely proportional to the square root of sample size. This guides how researchers plan studies to achieve desired precision.
Recommended video:
05:11Sampling Distribution of Sample Proportion
Related Videos
Related Practice