Unit: Sampling Distributions
Chapter: Biased & Unbiased Point Estimates
Reference: – Population & Sample, Point estimates & Parameters, Accuracy Bias, Unbiased point estimates, Interpreting & Comparing, Mean & Variance of sample means, Sample proportion & Bias, Maximum likelihood estimation, Methos of moments estimation, Sample size & estimation, Application & Examples.
After studying this chapter, you should be able to:
- Point estimates, Parameters & Bias in Point Estimates.
- Unbiased point estimates, Mean & Variance of Sample means.
- Sample Proportion & Maximum livelihood estimation.
- Method of Moments & Sample size estimation.
Point Estimates, Parameters & Bias in Point Estimates
Point Estimates:
- A point estimate is a single value that is used to approximate an unknown population parameter based on sample data.
- Point estimates provide a way to make educated guesses about population characteristics without having to observe the entire population.
- Common point estimates include the sample mean, sample proportion, and sample variance.
Parameters:
4. Parameters are numerical characteristics of a population that we aim to estimate using sample data.
- Examples of parameters include the population mean, population proportion, and population standard deviation.
- Parameters are typically fixed and unknown, making them the focus of statistical inference.
Bias in Point Estimates:
7. Bias refers to the systematic tendency of a point estimate to consistently deviate from the true population parameter.
- A point estimate is biased if, on average, it overestimates or underestimates the true parameter value.
- Bias can arise from the sampling method, measurement errors, or other sources of systematic error.
- Bias in point estimates can lead to inaccurate and misleading conclusions about the population.
Reducing Bias:
1. Unbiased point estimates are preferred because they, on average, provide accurate estimates of the population parameter.
- Techniques like random sampling and proper study design can help reduce bias in point estimates.
- Adjusting for bias involves using correction factors or more sophisticated statistical methods to obtain unbiased estimates.
Bias Correction Examples:
2. In estimating a population proportion, the sample proportion can be biased, but dividing by the correction factor (n-1) instead of n reduces bias in the estimate.
- In estimating population variance, using the Bessel's correction (n-1) instead of n corrects the bias in the sample variance.
Unbiased Point Estimates, Mean & Variance of Sample Means
Unbiased Point Estimates:
- An unbiased point estimate is a statistic that, on average, accurately estimates the population parameter it represents.
- Unbiasedness implies that if we repeatedly take random samples from a population and calculate the point estimate each time, the average of these estimates will be equal to the true population parameter.
- Unbiased point estimates are preferred because they do not systematically overestimate or underestimate the population parameter.
Mean of Sample Means:
- The mean of sample means, often denoted as "x̄" (x-bar), is the average of all possible sample means of a given sample size that can be drawn from a population.
- The mean of sample means is also referred to as the "expected value of the sample mean."
- According to the Central Limit Theorem (CLT), when sample sizes are sufficiently large (usually n ≥ 30), the distribution of sample means becomes approximately normal, regardless of the underlying population distribution.
Variance of Sample Means:
- The variance of sample means measures the spread or variability of the distribution of sample means around the population mean.
- The variance of sample means is influenced by two factors: the population variance (σ²) and the sample size (n).
- The formula for the variance of sample means is given by: Var(x̄) = σ² / n, where σ² is the population variance and n is the sample size.
- As the sample size increases, the variance of sample means decreases, leading to a more precise estimate of the population mean.
Sample Proportion & Maximum Livelihood Estimation
Sample Proportion:
- The sample proportion, denoted by "p̂" (p-hat), is a point estimate of the population proportion based on sample data.
- It represents the proportion of successes (or events of interest) in the sample.
- The sample proportion is used to estimate the population proportion, which is a parameter of interest.
- The sample proportion is calculated as the ratio of the number of successes to the total sample size.
Maximum Likelihood Estimation (MLE):
- Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model based on observed data.
- MLE aims to find the parameter values that maximize the likelihood function, which quantifies how well the model explains the observed data.
- In the context of sample proportion, MLE seeks the value of the population proportion that makes the observed sample outcomes most probable.
- MLE provides estimates that are efficient and asymptotically unbiased as the sample size increases.
Likelihood Function:
- The likelihood function is a probability distribution function that represents the probability of observing the given sample data for different values of the parameter.
- MLE involves finding the parameter value that maximizes the likelihood function, effectively making the observed data most likely under that parameter.
Applicability of MLE:
- MLE is widely used in various fields, including biology, economics, engineering, and social sciences, to estimate unknown parameters.
- MLE estimators are often preferred due to their desirable statistical properties, such as asymptotic efficiency.
Procedure for MLE:
- To perform MLE, formulate the likelihood function based on the observed data and parameter of interest.
- Take the derivative of the likelihood function with respect to the parameter and set it equal to zero to find the maximum.
- Solve for the parameter value that maximizes the likelihood function to obtain the MLE estimate.
Method of Moments & Sample Size Estimation
Method of Moments:
- The Method of Moments (MoM) is a statistical technique used to estimate the parameters of a population distribution based on moments of the sample data.
- Moments are mathematical measures of the shape and location of a distribution, such as mean, variance, skewness, and kurtosis.
- MoM seeks to equate the sample moments (usually up to a certain order) with the corresponding population moments and solve for the parameter estimates.
Procedure for Method of Moments:
- Identify a suitable mathematical model or distribution that describes the data.
- Express the population moments (e.g., mean, variance) in terms of the distribution's parameters.
- Equate the sample moments (calculated from the data) with the corresponding population moments and solve for the parameter estimates.
Advantages of Method of Moments:
- MoM provides a simple and intuitive way to estimate population parameters.
- It can be used even when complex statistical distributions are involved, provided the moments exist and are well-defined.
Limitations of Method of Moments:
- MoM may not always produce accurate estimates, especially for small sample sizes or when moments are poorly behaved.
- It may not perform well for distributions with heavy tails or highly skewed data.
Sample Size Estimation:
- Sample size estimation is the process of determining the number of observations needed in a sample to achieve a certain level of accuracy and confidence in statistical analysis.
- Adequate sample size is crucial for obtaining reliable and meaningful results in statistical inference.
Factors Influencing Sample Size:
Desired level of confidence (e.g., 95% confidence interval).
- Margin of error (precision) around the estimate.
Variability or expected standard deviation of the population.
Calculating Sample Size:
- Sample size calculations often involve formulas based on the desired level of confidence, margin of error, and variability.
- Software tools and statistical calculators are available to assist in sample size determination for various study designs and analyses.
Example: Estimating Average Income
Suppose you are conducting a survey to estimate the average income of households in a certain city. You randomly select a sample of 100 households and collect their income data. The true average income of all households in the city is $50,000.
Solution: – Biased Estimate:
Let's say that due to non-response bias, some higher-income households are less likely to participate in the survey. As a result, your sample tends to underrepresent high-income households. This leads to a biased estimate of the average income.
Suppose the average income in your sample is $48,000. This estimate is biased because it consistently underestimates the true average income due to the non-response bias.
Unbiased Estimate:
To correct for the bias, you can use a weighted approach. You know that the sample is biased towards lower incomes, so you can assign higher weights to the incomes of the higher-income households that did participate. This will help in adjusting the estimate to be closer to the true population average.
Suppose you calculate the weighted average income to be $49,000. This estimate is closer to the true average of $50,000 and is an unbiased estimate because, on average, it accurately represents the population parameter.
Key Points
- Definition: Point estimates are single values calculated from sample data to estimate unknown population parameters.
- Bias: Bias refers to a consistent tendency of a point estimate to systematically overestimate or underestimate the true population parameter.
- Biased Estimate: A biased estimate consistently deviates from the true parameter value in the same direction due to flaws in the sampling or measurement process.
- Unbiased Estimate: An unbiased estimate, on average, equals the true parameter value when considering all possible random samples.
- Selection Bias: Occurs when the method of selecting the sample systematically favors certain groups or characteristics, leading to an inaccurate estimate.
- Non-Response Bias: Arises when non-response by certain individuals or groups in a sample affects the estimate of the population parameter.
- Measurement Bias: Results from errors or inaccuracies in the measurement instrument or process used to collect data.
- Mitigating Bias:
- Weighted Estimates: Adjusting the contribution of each data point based on its perceived reliability can reduce bias in estimates.
- Random Sampling: Employing random sampling techniques can help mitigate bias by ensuring all individuals or groups have an equal chance of being included.
- Stratified Sampling: Dividing the population into subgroups and then randomly sampling from each subgroup can address bias and improve estimates.
- Unbiased Estimators:
- Definition: An estimator is unbiased if, on average, its expected value is equal to the true population parameter it aims to estimate.
- Sample Mean and Proportion: The sample mean and sample proportion are often unbiased estimators when calculated from random samples.
- Bias vs. Variability:
- Trade-off: While unbiasedness is desirable, it's important to balance bias and variability; reducing bias may increase variability.
- Efficiency: Biased estimators can sometimes have lower variability (greater precision) than unbiased estimators but may still lead to inaccurate results.
- Real-world Application:
- Election Polling Example: In election polling, biased samples can lead to inaccurate predictions, while unbiased samples are more likely to reflect the true voting patterns of the population.