Population/Sample Proportions

Overview

In statistics, a proportion refers to the fraction or percentage of a population or sample that exhibits a specific characteristic. Understanding proportions is key for making inferences about populations based on sample data.

Key Definitions

Population Proportion ($p$):

• Represents the proportion of the entire population that has a particular characteristic.
• It is often unknown and estimated using sample data.

Sample Proportion ($\hat{p}$):

• Represents the proportion of a sample with a specific characteristic.
• Calculated using:

$$\hat{p} = \frac{x}{n}$$

Where:

• $x$: Number of successes (or occurrences of the characteristic) in the sample.
• $n$: Total sample size.

Difference Between Population and Sample Proportions:

• Population proportion ($p$) is a parameter and remains constant.
• Sample proportion ($\hat{p}$) is a statistic and can vary between samples.

Using Sample Proportions to Estimate Population Proportions

Sample proportions are used to estimate population proportions through:

• Point Estimates: The sample proportion ($\hat{p}$) itself.
• Confidence Intervals: A range of values that likely contains the population proportion ($p$). The formula for a confidence interval for a population proportion is:

$$\hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Where:

• $z$: $Z-value$ for the desired confidence level.
• $\hat{p}$: Sample proportion.
• $n$: Sample size.

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Worked Examples

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Summary

• Population Proportion ($p$): The proportion of the entire population with a specific characteristic.
• Sample Proportion ($\hat{p}$): The proportion of a sample with the characteristic, calculated as:

$$\hat{p} = \frac{x}{n}$$

• Sample proportions are used to estimate population proportions using point estimates and confidence intervals.
• The formula for confidence intervals:

$$\hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

• Proportions are essential in inferential statistics to make predictions about populations.