When studying a population, we often rely on sample proportions. These proportions form a sampling distribution, which becomes normal as sample size grows. This distribution's center matches the population proportion, while its spread is measured by standard error.
Understanding the sampling distribution of proportions is crucial for making inferences about populations. It allows us to calculate confidence intervals and determine necessary sample sizes for accurate estimates. This knowledge is fundamental for statistical analysis in various fields.
Sampling Distribution of the Proportion
Sampling distribution of proportion
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Probability distribution of sample proportions obtained from repeated sampling of a population
Describes variability and behavior of sample proportions from different samples of the same size
Shape approaches a normal distribution as sample size increases, according to
True regardless of population distribution shape, if sample size is sufficiently large ([n](https://www.fiveableKeyTerm:n)≥30) and population is at least 10 times larger than sample
Center equals the population proportion ([p](https://www.fiveableKeyTerm:p))
Mean of sample proportions (μp^) is an unbiased estimator of population proportion
Spread measured by standard deviation, also known as (σp^)
Standard error decreases as sample size increases, indicating larger samples provide more precise estimates of population proportion
Standard error calculation
Standard error of the proportion (σp^) calculated using formula: σp^=np(1−p)
p is population proportion
n is sample size
Inversely related to sample size (n)
As sample size increases, standard error decreases, indicating larger samples provide more precise estimates of population proportion
Affected by population proportion (p)
When p is close to 0 or 1, standard error is smaller compared to when p is close to 0.5, assuming constant sample size
Confidence intervals for proportions
Range of values likely to contain true population proportion with specified level of confidence
Constructed using formula: p^±zα/2⋅σp^
p^ is sample proportion
zα/2 is critical value from standard normal distribution corresponding to desired confidence level
σp^ is standard error of the proportion
Interpreted as: "We are (1−α)% confident that the true population proportion falls within the calculated interval"
95% means if we repeatedly sample population and construct intervals, about 95% would contain true population proportion
Sample size determination
Minimum sample size required depends on desired level of confidence, , and estimate of population proportion
Calculated using formula: n=E2zα/22⋅p^(1−p^)
zα/2 is critical value from standard normal distribution corresponding to desired confidence level
p^ is estimate of population proportion (often 0.5 if no prior information available)
E is desired margin of error
If calculated sample size is more than 5% of population size, use finite population correction factor to adjust: nadjusted=1+Nn−1n