Statistical Estimation | UGC NET Commerce Preparation Course PDF Download
| Table of contents |
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| What is Estimation? |
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| Purpose of Estimation in Statistics |
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| Types of Estimation |
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| Estimation Methods |
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| Estimators as Random Variables |
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| Factors Affecting Estimation |
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What is Estimation?
Estimation in statistics involves using sample data to make informed guesses about a population’s characteristics, such as the mean, variance, or proportion. Since measuring every member of a population is often impractical, statisticians use sample data to infer population parameters.
Purpose of Estimation in Statistics
Statistical estimation is crucial for making inferences about populations based on sample data. It enables the estimation of parameters like the mean and variance without measuring each individual in the population. Estimation plays a key role in decision-making across various fields, including business and healthcare, guiding strategies, treatment plans, and risk management in finance and insurance. It also supports quality control by ensuring that products and services meet established standards, allowing for the detection and correction of deviations.
Types of Estimation
There are two main types of estimation:
- Point Estimation
- Interval Estimation
Point Estimation:
- Point estimation involves using a single value, such as the sample mean, to estimate a population parameter. For example, measuring the heights of a few individuals and calculating the average height provides an estimate for the average height of the entire population. Although point estimates may not be precise, they offer a useful approximation of population characteristics, such as proportions or means.
Interval Estimation:
- Unlike point estimation, which provides a single value, interval estimation gives a range of values within which the true population parameter is likely to fall. This approach accounts for data variability and the inherent uncertainty in estimation. For instance, when estimating the number of jelly beans in a jar, offering a range like 80 to 120 beans (a confidence interval) is more informative than giving a single number. Confidence intervals help account for potential errors in the estimation process and provide a more flexible understanding of the true parameter.
Examples of Estimation in Statistics
- Population Mean Estimation: To estimate the average height of adults in a country, a random sample of adults is taken, and the sample mean is calculated as an estimate of the population mean.
- Population Proportion Estimation: To gauge the percentage of supporters for a political candidate, a random sample of voters is surveyed, and the sample proportion is used as an estimate of the population proportion.
- Interval Estimation for Mean: By calculating a confidence interval using the sample mean and standard deviation, one can estimate the average transaction time in a store, providing a likely range for the population mean.
- Regression Analysis: This method estimates relationships between variables, such as income and education level, by fitting a model to data and estimating coefficients that describe the population relationship.
- Bayesian Estimation: This approach combines prior knowledge with current data to update beliefs about a parameter, such as the effectiveness of a drug treatment.
Estimation Methods
Several techniques can be used to generate estimates in statistics:
- Method of Moments
- Maximum Likelihood Estimation (MLE)
- The method of moments involves comparing moments (such as measures of central tendency and spread) computed from sample data with the corresponding moments in the population. By solving the resulting equations, population parameters can be estimated.
- For example, the method of moments can estimate the mean age of a large population by analyzing the age distribution in a sample subset. By calculating moments for the sample, the group’s mean and age distribution can be determined, and these estimates are extended to infer the characteristics of the larger population. This technique enables the estimation of population traits based on smaller sample data and provides reliable results for larger populations.
- Maximum likelihood estimation (MLE) seeks to find parameter values that maximize the likelihood of observing the given data within a statistical model. By adjusting the parameters iteratively, the method aims to identify values that make the observed data most probable.
- MLE starts with an initial estimate for a parameter and then refines it through iterative adjustments to maximize the likelihood of the observed data. By comparing various estimates to the dataset, MLE helps pinpoint the parameter values that best fit the data. This approach is particularly useful for accurately estimating unknown variables by optimizing the likelihood of the observed outcomes.
Estimators as Random Variables
Estimators are considered random variables in statistics because they are computed from random samples, leading to variation in their values.
- Sample Variability: When drawing samples from a population, estimators like the sample mean will vary between different samples.
- Sampling Distribution: The sampling distribution of an estimator represents the range of values it can take when calculated from different samples of a given size, offering insights into its characteristics and variability.
- Bias and Variance: An estimator may have bias, consistently over- or underestimating the true parameter. Variance measures how much the estimator values spread around its predicted value. Both bias and variance affect the estimator’s accuracy.
- Mean and Variance of Estimators: Just like random variables, estimators have a mean and variance. The mean of an estimator should equal the parameter being estimated, while the variance reflects its precision.
- Efficiency and Consistency: Efficiency refers to the accuracy of an estimator in estimating a parameter with sample data, with lower variance indicating greater efficiency. Consistency means the estimator converges toward the true parameter value as sample size increases.
- Central Limit Theorem: According to the Central Limit Theorem, regardless of the population distribution, the sampling distribution of many estimators approaches a normal distribution as sample size increases. Understanding this theorem is essential to comprehend estimator behavior.
Statisticians use statistical techniques and probability theory to evaluate the performance, characteristics, and reliability of estimators, treating them as random variables in their analysis.
Factors Affecting Estimation
Several factors influence the accuracy of statistical estimation:
Sample Size: Larger sample sizes result in more precise estimates, increasing the likelihood of accurately representing the population parameter.
For example, estimating the average height of students in a school is more accurate with a larger sample size. Measuring only five students may not provide a reliable estimate, but measuring 50 or 500 students will give a better understanding of the true average height. In short, analyzing more individuals leads to a more accurate representation of the population's characteristics.
Sampling Method: The sampling method significantly impacts the accuracy of estimates. A random sampling approach, where every individual in the population has an equal chance of being selected, helps ensure unbiased estimates, thereby improving accuracy.
Random sampling is essential for producing accurate estimations. By selecting individuals purely by chance, this method ensures that the sample fairly represents the entire population. Whether determining the distribution of colored candies in a jar or identifying the most popular ice cream flavor in a town, random sampling helps avoid bias, leading to more reliable conclusions about the population or issue being studied.
Solved Problems
Problem 1: 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.
Sol:
We can use the sample mean (average) as a point estimate for the population mean.
Sample Mean (x̄) = Σ(xi) / n
where:
Σ (sigma) represents the sum
xi 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: Continuing from Problem 1, suppose we want to estimate the average score for the entire class with a 95% confidence level.
Sol:
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.
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FAQs on Statistical Estimation - UGC NET Commerce Preparation Course
| 1. What are some common statistical estimation techniques used in data analysis? |  |
| 2. Why is understanding estimation important in decision-making processes? | |
| 3. How does sample size impact the accuracy of statistical estimation? | |
| 4. What are key statistical concepts that are essential for effective estimation in statistics? | |
| 5. How does Bayesian estimation differ from traditional frequentist estimation methods? | |
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Nov 24, 2024 Last updated |
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