14: Population Estimation

14.1 Introduction to Population Estimation

Population estimation involves estimating population parameters, such as the mean or proportion, based on a sample drawn from that population. Since collecting data from an entire population is often impractical or impossible, researchers rely on samples to make informed inferences about the larger population. While population estimation is part of inferential statistics, it begins with descriptive statistics, summarizing sample data to make predictions or estimates about the population. In this sense, population estimation acts as a bridge between descriptive and inferential statistics.

In this chapter, we will cover estimating population means and proportions, constructing confidence intervals to quantify uncertainty in estimates, and understanding sampling error and standard error and how they impact the accuracy of our estimates. Population estimation is essential for predicting a population based on sample data, providing a quantifiable confidence level in the findings.

14.2 Estimating Population Parameters

We use sample statistics to estimate population parameters. The most commonly estimated parameters are the population mean and population proportion.

Estimating the Population Mean

The sample mean (average) is typically used to estimate the population mean. If the sample is randomly selected and sufficiently representative, the sample mean approximates the population mean.

Estimating the Population Proportion

The sample proportion is used to estimate the population proportion for categorical data. This is calculated by dividing the number of successes (or occurrences of a particular characteristic) by the total sample size.

14.3 Sampling Error and Its Impact

Sampling error is the difference between the sample statistic (the sample mean or proportion) and the true population parameter. This error arises because a sample is just a subset of the population and may not represent the entire population perfectly. Larger samples tend to reduce sampling error, providing a more accurate estimate of the population parameter. Random sampling ensures that every individual in the population has an equal chance of being selected, which minimizes bias and increases the sample’s representativeness.

Understanding sampling error is important because it helps us recognize the uncertainty in our estimates. This is why techniques like confidence intervals are used to quantify the potential error in our estimates.

14.4 Standard Error

Standard error (SE) measures the variability of a sample statistic (such as the sample mean or sample proportion). It quantifies how much the statistic is expected to fluctuate from sample to sample. Standard error is essential in constructing confidence intervals and assessing the precision of estimates.

Standard error helps assess the accuracy of a sample statistic as an estimate of the population parameter. It decreases as the sample size increases, which means larger samples provide more precise estimates. Standard error is used to calculate confidence intervals around sample estimates. For the mean, the standard error is calculated by dividing the sample’s standard deviation (s) by the square root of the sample size (n). For proportions, it is calculated using the sample proportion (p̂).

14.5 Confidence Intervals

A confidence interval (CI) is a range of values likely to contain the true population parameter based on the sample data. It provides a way to express the uncertainty around an estimate, and a higher confidence level gives you more certainty that the interval contains the true value. For example, a 95% confidence interval means that if you were to take 100 different samples from the same population, 95 of the intervals you calculate would contain the true population parameter.

Confidence intervals are important because they allow you to express the uncertainty in your estimates. They provide a range of plausible values for the population parameter, helping you understand the precision of your estimate. Wider intervals suggest less precision, while narrower intervals indicate more confidence in the estimate.

The confidence level is the probability that the confidence interval contains the true population parameter. Common confidence levels are:

• 90% confidence (provides a slightly wider range but gives less certainty).
• 95% confidence (the most commonly used level, providing a good balance between certainty and precision).
• 99% confidence (a wider range, offering more certainty but less precision).

A higher confidence level results in a wider confidence interval. While this increases the certainty that the interval contains the true value, it also reduces the estimate’s precision.

14.6 Interpreting Confidence Intervals

Interpreting confidence intervals is an essential skill for making inferences about populations. A confidence interval provides a range of values for the population parameter, and the width of the interval indicates how precise the estimate is. For example, suppose your sample mean is 50, and the 95% confidence interval is (47, 53). This means you can be 95% confident that the true population mean lies between 47 and 53. If you were to take 100 different samples from the same population, 95% of the resulting confidence intervals would contain the true population mean.

14.7 Population Estimation in Jamovi

Jamovi makes it easy to estimate population and construct confidence intervals through its built-in Exploration menu. To estimate population parameters in Jamovi, open your dataset and go to the Exploration menu under Analyses. Select the variable you want to analyze, such as test scores or a proportion of successes. In the Statistics options, select Mean or Proportion depending on the type of data you are analyzing. Then, check the box for Confidence Interval to generate the confidence interval for the mean or proportion. Choose the desired confidence level (e.g., 95%), and click OK to generate the output. Jamovi will display the point estimate (mean or proportion) and the confidence interval, allowing you to see the range within which the true population parameter will likely fall.

Below is an example of the results generated when the steps are correctly followed.

IMAGE [INSERT NAME OF DATASET]

Interpretation