Estimation in Statistics

Estimation is a technique for calculating information about a bigger group from a smaller sample, and statistics are crucial to analyzing data. For instance, the average age of a city’s population may be obtained by taking the age of a sample of 1,000 residents. While estimates aren’t perfect, they are typically trustworthy enough to be of value.

In this article, we examine the significance of statistics, their function in the interpretation of data, and how efficient data analysis leads to the making of decisions based on accurate information.

Estimation in statistics involves using sample data to make educated guesses about a population’s characteristics, such as mean, variance, or proportion. The population refers to the entire interest group, like all people in a country or all products made by a company.

Since it’s often impractical to measure every member of a population, statisticians rely on samples to make inferences about the entire population. Estimation helps to conclude population parameters based on sample data.

Statistical estimation is essential for making inferences about populations using sample data, helping to determine parameters like mean and variance without individual measurements.

• This evaluation is vital for decision-making in business and healthcare, informing strategies and treatment options.
• It is closely linked to hypothesis testing, contributing to scientific development, political decisions, public health, and economic choices.
• Risk assessment benefits from evaluation in managing probabilities and risk in finance and insurance.
• Quality control also relies on evaluation to ensure products and services meet standards by identifying and correcting deviations.

Estimation is of two types that includes:

• Point Estimation
• Interval Estimation

Point Estimation

In statistics, the sample mean is used to estimate a population mean, while the sample proportion is used to estimate a population percentage. These measurements help approximate unknown population parameters accurately.

• Identifying a single number to represent a large group is like a point estimate. For instance, measuring the heights of random people can be used to estimate the average height of the entire group.

If individuals measured were 5 feet, 6 feet, and 5 feet.

We could estimate the average height to be around 5 feet. This single number, called a score estimator, gives a rough idea of the group’s characteristics.

• Population mean is estimated using the sample mean.
• Similar techniques can be applied to estimate other attributes like percentages of specific characteristics in a population.

While not always precise, these estimates offer a good understanding of the group’s traits.

Interval Estimation

Point estimates provide a single value, while interval estimates give a range likely to contain the true parameter. This method recognizes data variability and estimation uncertainty.

When estimating the number of jelly beans in a jar, it is better to provide a range, known as a confidence interval, rather than a single guess. This range, such as 80 to 120 jelly beans, allows for uncertainty in the estimate and acknowledges the margin of error.

Confidence intervals give us a sense of freedom in our estimations, while point estimates only provide a single number without considering this uncertainty.

It helps us understand that there is some level of uncertainty in the estimation process.

Some examples of estimation in statistics:

• Population Mean Estimation: To estimate the average height of all adults in a country, take a random sample of adults and calculate the sample mean as an estimate of the population mean height.

• Population Proportion Estimation: To estimate the percentage of supporters for a candidate in a city, survey a random sample of voters and use the sample proportion as an estimate for the population proportion.

• Interval Estimation for Mean: Estimate average transaction time in a store by calculating confidence interval using sample mean and standard deviation to determine likely range of population mean event time.

• Regression Analysis: Regression analysis estimates the relationship between variables, such as income and education level. By fitting a model to data, coefficients describing the population relationship can be estimated.

• Bayesian Estimation: Bayesian estimation integrates prior knowledge with current data to update beliefs on parameters of interest, such as evaluating drug treatment effectiveness.

There are several techniques that can be used to generate estimates:

• Method of Moments
• Maximum Likelihood Estimation (MLE)

1. Method of Moments

This method compares the moments (central tendency and spread) that are computed from the sample data to the corresponding moments in the population. The population parameters can be estimated by working out the resulting equations.

• The approach of moments can estimate the mean age of a large population by analyzing age distribution. Moments are calculated for a subset to determine group mean and age distribution, which is then used to estimate the larger population. Trained estimates can be made by comparing moments of smaller and larger groups, allowing inference of population characteristics based on smaller sample data. This method provides accurate estimates for larger populations using smaller sample data.

2. Maximum Likelihood Estimation (MLE)

Maximum likelihood estimation (MLE) aims to find parameter values that give the highest chance of observing the data in a statistical model. It involves identifying values that maximize the likelihood of the observed data.

• Maximum likelihood estimation (MLE) is a method used to find the most probable values of variables based on given data. It involves starting with an initial estimate for a parameter and iteratively adjusting it to maximize the likelihood of observing the data. By comparing different estimates to the dataset, the MLE process helps identify the parameter values that best fit the data. This statistical method is valuable in accurately estimating unknown variables by increasing the probability of occurrence in the dataset through adjusting parameter values in a model.

An estimator in statistics is considered a random variable as it’s computed from random data samples, leading to varying values.

• Sample Variability: When sampling from a population, we randomly select a subset of people or observations. Estimators such as sample mean vary between samples.

• Sampling Distribution: The sampling distribution of an estimator shows the potential values it can take when calculated from various samples of a certain size from a population, providing insights into its characteristics and variability.

• Bias and Variance: Estimators can have bias, consistently overestimating or underestimating the true parameter. Variance measures the spread of estimator values around its predicted value. Both variance and bias impact the accuracy of estimators.

• Mean and Variance of Estimators: Estimators have the same mean and variance as random variables. The mean of an estimator should be equal to the parameter it is estimating. The variance of an estimator indicates its precision or variability.

• Efficiency and Consistency: Efficiency measures an estimator’s accuracy in estimating a parameter with sample data. A smaller variance indicates better efficiency. Consistency means the estimator approaches the correct parameter value as sample size increases.

• Central Limit Theorem: Central Limit Theorem states that regardless of population distribution, the sampling distribution of many estimators becomes normal as sample size increases. Understanding this theorem is essential to grasp the behavior of estimators.

Statisticians analyze estimator performance, characteristics, and reliability using statistical techniques and probability theory with random variables.

Various factors affecting estimation are:

1. Sample Size: Larger sample sizes lead to more precise estimates, increasing the likelihood of accurately representing the population parameter.

• Estimating the average height of students in a school is more accurate with a larger sample size. Measuring just five students may not be reliable, but measuring 50 or even 500 students can provide a better idea of the true average height. A larger sample size leads to a more accurate estimate of the entire population’s characteristics. In short, studying more individuals results in a more precise estimate of the entire population.

2. Sampling Method: The sampling method affects estimate accuracy. A random sample with every member having an equal chance ensures an unbiased estimate, improving accuracy.

• Sampling method is crucial for accurate estimations.
• Random sampling selects individuals purely by chance, giving each an equal chance of being chosen.
• This ensures a fair representation of the entire group, making it useful for determining distributions like colored candies in a jar or favorite ice cream flavors in a town without bias.
• Random sampling helps reflect the opinions of the whole group, not just a subset, leading to fair and unbiased findings for drawing accurate conclusions about a population or problem.

Problem 1: Point Estimation (Mean)

A random sample of 10 students from a college class scored the following marks in an exam: 85, 78, 92, 80, 65, 90, 72, 88, 95, 83. Estimate the average score for the entire class.

Solution:

We can use the sample mean (average) as a point estimate for the population mean.

Sample Mean (x̄) = Σ(x_i) / n

where:

Σ (sigma) represents the sum

x_i represents the individual score of each student (i = 1 to 10)

n is the sample size (n = 10)

Calculating the sum of scores: Σ(xi)= 833

Therefore, Sample Mean (x̄) = 833 / 10 = 83.3

Interpretation: Based on this sample, we can estimate the average score for the entire class to be around 83.3.

Problem 2: Point Estimation (Proportion)

A survey of 200 customers at a grocery store revealed that 120 prefer brand A. Estimate the proportion of all customers who prefer brand A.

Solution:

We can use the sample proportion (p̂) as a point estimate for the population proportion (p).

Sample Proportion (p̂) = x / n

Therefore, Sample Proportion (p̂) = 120 / 200 = 0.6

Interpretation: Based on the survey, we can estimate that around 60% of all customers at the store prefer brand A.

Problem 3: Interval Estimation (Confidence Interval for Mean)

Continuing from Problem 1, suppose we want to estimate the average score for the entire class with a 95% confidence level.

Solution:

When a 95% confidence level with 9 degrees of freedom is used, the t-critical value (t*) must be calculated in order to express a mean value using a confidence interval.

Statistical software or internet tables can be used to find T*.

Assuming t* = 2.262 (for a 95% CI with 9 degrees of freedom), we can calculate the margin of error (ME):

ME = (t*) × (standard deviation / √n)

Since we don’t have the population standard deviation, we can estimate it using the sample standard deviation (s). You’ll need to calculate the sample standard deviation for the scores (around 5.8).

Example Calculation (assuming s = 5.8):

ME = (2.262) × (5.8 / √10) ≈ 4.3

Now, we can construct the confidence interval:

CI = Sample Mean (x̄) ± ME

CI = 83.3 ± 4.3

Therefore, the 95% confidence interval for the mean score is approximately (79, 87.6).

Interpretation: We can be 95% confident that the true average score for the entire class lies somewhere between 79 and 87.6.

Problem 4: Interval Estimation (Confidence Interval for Proportion)

Following the grocery store survey (Problem 2), construct a 90% confidence interval for the proportion of customers who prefer brand A.

Solution:

Similar to the previous problem, we can use the confidence interval for a proportion. We’ll need to find the z-critical value (z*) for a 90% confidence level (typically 1.645).

Calculation:

CI = p̂ ± (z*) × √(p̂(1-p̂) / n)

CI = 0.6 ± (1.645) × √(0.6 × (1-0.6) / 200) ≈ 0.6 ± 0.05

Therefore, the 90% confidence interval for the proportion who prefer brand A is approximately (0.55, 0.65).

Interpretation: We can be 90% confident that the true proportion of customers who prefer brand A in the entire population falls between 55% and 65%.

What is the difference between point estimation and interval estimation?

In statistics, point estimates and interval estimates are two primary methods used to estimate population parameters from sample data.

• A point estimate provides a single value as the best estimate for an unknown parameter, while an interval estimate creates a range of values, known as a confidence interval, that are likely to include the true parameter at a particular confidence level.
• Interval estimates consider the estimate’s uncertainty, while point estimates offer a single figure representing the most likely result.

How do we choose the right estimator for a particular problem?

Selecting an appropriate estimator is not a universally applicable approach.

• It is essential to consider the type of data, its distribution, and the desired features of estimation when selecting an appropriate estimator.
• Analyzing facts, prioritizing, investigating potential estimates, and comparing them will lead to an informed conclusion.

What factors can affect the width of a confidence interval?

• Sample Size: Larger sample sizes generally result in narrower confidence intervals.
• Variability of Data: Higher variability typically leads to wider confidence intervals.
• Confidence Level: Higher confidence levels result in wider confidence intervals.
• Choice of Estimator: Different estimators may result in different widths of confidence intervals.

How can we assess the accuracy of an estimation method?

Compare the estimated values to known or true values if available. Use statistical measures such as mean squared error or bias to quantify the accuracy of estimates.

What are the limitations of estimation in statistics?

Error is a common problem in estimating since it is a judgment call based on sample data on population characteristics.

• Estimate may not reflect the real population distribution that it expects.
• Extreme values or data outliers may cause estimation systems to become sensitive.

A few examples of variables that may have an impact on an estimate’s accuracy are sample size, sampling strategy, and data variability.