Have you ever wondered how statisticians determine parameters such as the mean age of an entire country's population? It's impossible to gather data from every single member of the population, so they rely on small samples to estimate these statistics.
This process is known as point estimation. In this article, we will explore the definition of point estimation, various estimation methods, and their corresponding formulas, along with some examples.
Definition of Point Estimation
Before delving into point estimation, let's review some key concepts: population, sample, parameter, and statistic.
- The population is the group under study, from which statistical inferences are drawn;
- A parameter is a numerical characteristic of the population;
- A sample is a representative subset of the population;
- A statistic is a numerical characteristic of the sample.
Point estimation involves using statistics from one or more samples to estimate an unknown parameter of a population.
Researchers often lack knowledge of population parameters, highlighting the importance of representative samples in statistical studies.
Formulas for Point Estimation
Various population parameters have different estimators with unique formulas for estimation. Let's explore some common terminologies and notations.
The result of a point estimation is a single value, known as the estimator, typically denoted with a hat '^'.
Refer to the table below for examples of estimators, parameters, and their respective notations.
|
Parameter |
Notation |
Point Estimate |
Notation |
|
Mean |
\(\mu\) |
Sample mean |
\(\hat{\mu}\) or \(\bar{x}\) |
|
Proportion |
\(p\) |
Sample proportion |
\(\hat{p}\) |
|
Variance |
\(\sigma^2\) |
Sample variance |
\(\hat{s}^2\) or \(s^2\) |
Methods of Point Estimation
There are various methods of point estimation, such as maximum likelihood, least squares, and best-unbiased estimator, each ensuring credible estimators with specific properties.
- Consistent: It is important to have a large sample size to ensure the accuracy of the estimator value;
- Unbiased: The estimators of samples drawn from the population should closely approximate the true value of the population parameter (with a small standard error).
The estimators presented in the previous table are unbiased in estimating the parameters. To delve deeper into this topic, refer to our article on Biased and Unbiased Point Estimates.
When an estimator satisfies the above two properties, it is considered the most efficient or best-unbiased estimator. Among consistent, unbiased estimators, the preference is for the most consistent and unbiased one.
Next, familiarize yourself with two essential estimators: the and the estimator for the proportion. These are the best-unbiased estimators for their respective parameters.
Point Estimate of the Mean
Let's begin with the first estimator, the sample mean, \(\bar{x}\), of the population mean, \(\mu\). Its formula is
\[\bar{x}=\frac{\sum\limits_{i=1}^{n}x_i}{n},\]
where
- \(x_i\) represents the data points (observations) of a sample;
- \(n\) denotes the sample size.
After understanding the concept, we can see that the best unbiased estimator of the population mean is based on the arithmetic mean.
Let's illustrate this with an example:
Given the values below, we need to find the best point estimate for the population mean \(\mu\).
\[7.61, 7.17, 9.06, 6.305, 7.805, 7.11, 9.705, 6.11,8.56, 7.11, 6.455, 9.06\]
Solution:
To find the best point estimate, we calculate the sample mean of this data.
\[\begin{align} \bar{x}&=\frac{\sum\limits_{i=1}^{n}x_i}{n} \\ &= \frac{7.61+7.17+9.06+6.305+7.805+7.11+9.705+6.11+8.56+7.11+6.455+9.06}{12} \\ &= 7.67 \end{align} \]
Therefore, the best point estimate for the population mean \(\mu\) is \(\bar{x}=7.67\).
Another estimator related to the mean is the difference between two means, denoted as \( \bar{x}_1-\bar{x}_2\). This estimator is useful for comparing the same characteristic between two populations.
Point Estimate of Proportion
The population proportion can be estimated by dividing the number of successes in the sample \(x\) by the sample size (n). This can be expressed as:
\[ \hat{p}=\frac{x}{n}\]
When calculating the proportion of a specific characteristic, each element in the sample that possesses that characteristic is considered a success.
Let's consider an example:
A survey was conducted with \(300\) teacher trainees, where \(103\) of them viewed the services favorably out of a total of \(150\) respondents. The point estimation for this data is:
\[ \hat{p} = {103\over 150} = 0.686\]
Therefore, the sample proportion is \(0.686\) or \(68.7\%\).
When comparing proportions of two populations, one useful estimator is the difference of two proportions, denoted as \( \hat{p}_1-\hat{p}_2\). This estimator can be helpful when trying to determine if one population has a significantly different proportion compared to another. For example, if you suspect that one of two coins is biased because it consistently lands on heads more often, this estimator can help quantify that difference.
Example of Point Estimation
Point estimation involves several key components:
- Data: The information gathered from a sample is essential for estimation.
- Unknown parameter: This is the value that needs to be estimated for the entire population.
- Estimator formula: A formula is used to estimate the unknown parameter.
- Estimator value: The calculated value based on the sample data.
It's important to consider all these elements when working on point estimation problems.
For instance, a researcher wants to estimate the proportion of students who visit their college library at least three times a week. They surveyed \(200\) science students, with \(130\) visiting the library at least \(3\) times, and \(300\) humanities students, with \(190\) meeting the same criteria.
a) Calculate the proportion of science students visiting the library at least \(3\) times a week.
b) Determine the proportion of humanities students meeting the same criteria.
c) Identify which group visits the library more frequently.
Solution:
a) Let \(x\) be the number of science students visiting the library at least \(3\) times a week, so \(x=130\); and \(n=200.\) For the science group,
\[\hat{p}=\frac{130}{200}=0.65.\]
b) Let \(x\) be the number of humanities students visiting the library at least \(3\) times a week, so \(x=190\); and \(n=300.\) For the humanities group,
\[\hat{p}=\frac{190}{300}=0.63.\]
c) The proportion of science students visiting the library is higher than that of humanities students. Therefore, it can be concluded that more science students visit the library frequently.
Point Estimation vs. Interval Estimation
After reading this article, you may have realized that point estimation provides a numerical approximation of the population parameter you are interested in. However, the drawback of this method is the uncertainty regarding how close or far the estimator is from the true parameter value. This is where interval estimation comes into play, taking into account the margin of error to gauge the estimator's proximity to the parameter.
It is crucial for the estimated parameter values to be as accurate as possible, as this enhances the credibility of statistical inferences.
For more information on interval estimation, refer to the article .
Key Takeaways on Point Estimation
- Point estimation involves using statistics from one or more samples to estimate an unknown population parameter.
- Two crucial properties of estimators are
- Consistency: the accuracy of the estimator improves with larger sample sizes;
- Unbiasedness: the estimators' values should closely align with the true population parameter value.
- When an estimator satisfies these properties, it is considered the best-unbiased estimator.
- The best-unbiased estimator for population mean \(\mu\) is the sample mean \(\bar{x}\) calculated using the formula\[\bar{x}=\frac{\sum\limits_{i=1}^{n}x_i}{n}.\]
- The best-unbiased estimator for population proportion \(\mu\) is the sample proportion \(\hat{p}\) determined by the formula\[\hat{p}=\frac{x}{n}.\]
- A limitation of point estimation is the uncertainty surrounding the estimator's proximity to the true parameter value, highlighting the importance of interval estimation.