Textbook Question
True or False: The population proportion and sample proportion always have the same value.
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8. Sampling Distributions & Confidence Intervals: Proportion
Sampling Distribution of Sample Proportion
3:02 minutesProblem 8.2.6
Textbook Question
What happens to the standard deviation of p̂ as the sample size increases? If the sample size is increased by a factor of 4, what happens to the standard deviation of p̂?
Verified step by step guidance1
Recall that the standard deviation of the sample proportion \(\hat{p}\), often called the standard error, is given by the formula:
\[\text{SD}(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}\]
where \(p\) is the true population proportion and \(n\) is the sample size.
Notice from the formula that the standard deviation of \(\hat{p}\) depends on the sample size \(n\) in the denominator inside the square root. This means that as \(n\) increases, the denominator gets larger, making the whole fraction smaller, and thus the standard deviation decreases.
To understand the effect of increasing the sample size by a factor of 4, replace \(n\) with \$4n$ in the formula:
\[\text{SD}_{new} = \sqrt{\frac{p(1-p)}{4n}}\]
Simplify the expression by factoring out the 4 inside the square root:
\[\text{SD}_{new} = \sqrt{\frac{1}{4}} \times \sqrt{\frac{p(1-p)}{n}} = \frac{1}{2} \times \text{SD}(\hat{p})\]
This shows that increasing the sample size by a factor of 4 reduces the standard deviation of \(\hat{p}\) by a factor of 2, meaning the estimate becomes more precise as the sample size grows.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Deviation of a Sample Proportion
The standard deviation of the sample proportion (p̂) measures the variability of p̂ around the true population proportion (p). It is calculated as the square root of [p(1 - p) / n], where n is the sample size. This quantifies how much p̂ is expected to fluctuate from sample to sample.
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Effect of Sample Size on Variability
As the sample size (n) increases, the variability or standard deviation of the sample proportion decreases. This is because a larger sample provides more information, leading to more precise estimates and less fluctuation in p̂.
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Impact of Increasing Sample Size by a Factor
When the sample size is increased by a factor of 4, the standard deviation of p̂ decreases by a factor of 2, since standard deviation is inversely proportional to the square root of the sample size. This means doubling the sample size reduces variability by about 29%, quadrupling reduces it by half.
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