Estimates and Estimation

Post about Estimates and Estimation that include point estimation, interval estimation, level of significance, Properties of estimators, and real-life examples about estimate and estimation.

Statistical Inference: An Introduction

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Introduction to Statistical Inference

Inference means conclusion. When we discuss statistical inference, it is the branch of Statistics that deals with the methods to make conclusions (inferences) about a population (called reference population or target population), based on sample information. The statistical inference is also known as inferential statistics. As we know, there are two branches of Statistics: descriptive and inferential.

Statistical inference is a cornerstone of many fields of life. It allows the researchers to make informed decisions based on data, even when they can not study the entire population of interest. The statistical inference has two fields of study:

Statistical Inference

Estimation

Estimation is the procedure by which we obtain an estimate of the true but unknown value of a population parameter by using the sample information that is taken from that population. For example, we can find the mean of a population by computing the mean of a sample drawn from that population.

Estimator

The estimator is a statistic (Rule or formula) whose calculated values are used to estimate (a wise guess from data information) is used to estimate a population parameter $\theta$.

Estimate

An estimate is a particular realization of an estimator $\hat{\theta}$. It is the notation of a sample statistic.

Types of Estimators

An estimator can be classified either as a point estimate or an interval estimate.

Point Estimate

A point estimate is a single number that can be regarded as the most plausible value of the $\theta$ (notation for a population parameter).

Interval Estimate

An interval estimate is a set of values indicating confidence that the interval will contain the true value of the population parameter $\theta$.

Testing of Hypothesis

Testing of Hypothesis is a procedure that enables us to decide, based on information obtained by sampling procedure whether to accept or reject a specific statement or hypothesis regarding the value of a parameter in a Statistical problem.

Note that since we rely on samples, there is always some chance our inferences are not perfect. Statistical inference acknowledges this by incorporating concepts like probability and confidence intervals. These help us quantify the uncertainty in our estimates and test results.

Important Considerations about Testing of Hypothesis

  • Hypothesis testing does not prove anything; it provides evidence for or against a claim.
  • There is always a chance of making errors (Type I or Type II).
  • The results are specific to the chosen sample and significance level.

Statistical Inference in Real-Life

Some real-life examples of inferential statistics:

  1. Medical Trials: When a new drug is developed, it is tested on a sample of patients to infer its effectiveness and safety for the general population. Statistical inference helps determine whether the observed effects are due to the drug or random chance.
  2. Market Research: Companies use inferential statistics to understand consumer preferences and behaviours. By surveying a sample of consumers, they can infer the preferences of the broader market and make informed decisions about product development and marketing strategies.
  3. Public Health: Epidemiologists use statistical inference to track the spread of diseases and the effectiveness of interventions. Analyzing sample data one can infer the overall impact of a disease and the effectiveness of measures like vaccinations.
  4. Quality Control: Manufacturers use statistical inference to monitor product quality. By sampling a few items from a production batch, they can infer the quality of the entire batch and make decisions about whether to continue production or make adjustments.
  5. Election Polling: Pollsters use samples of voter opinions to infer the likely outcome of an election. Statistical inference helps estimate the proportion of the population that supports each candidate and the margin of error in these estimates.
  6. Education: Educators and policymakers use statistical inference to evaluate the effectiveness of teaching methods and educational programs. By analyzing test scores and other performance metrics from a sample of students, they can infer the impact of these methods on the broader student population.
  7. Environmental Studies: Researchers use statistical inference to assess environmental impacts. For example, by sampling air or water quality in specific locations, they can infer the overall environmental conditions and the effectiveness of pollution control measures.
  8. Sports Analytics: Teams and coaches use statistical inference to evaluate player performance and strategy effectiveness. By analyzing data from a sample of games, they can infer the overall performance trends and make decisions about training and game strategy.
  9. Finance: Investors and financial analysts use statistical inference to make decisions about investments. By analyzing sampled historical data of stocks or other financial instruments, one can infer future performance and make informed investment decisions.
  10. Customer Satisfaction: Businesses use statistical inference to gauge customer satisfaction and loyalty. By surveying a sample of customers, one can infer the overall satisfaction levels and identify areas for improvement.

FAQs about Statistical Inference

  1. Define the term estimation.
  2. Define the term estimate.
  3. Define the term estimator.
  4. Write a short note on statistical inference.
  5. What is statistical hypothesis testing?
  6. What is the estimation in statistics?
  7. What are the types of estimations?
  8. Write about point estimation and intervention estimation.

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Estimation Online Quiz

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MCQs from Statistical Inference covering the topics of Estimation and Hypothesis Testing for the preparation of exams and different statistical job tests in Government/ Semi-Government or Private Organization sectors. This Estimation online quiz will also help get admission to different colleges and Universities. The Estimation Online Quiz will help the learner to understand the related concepts and enhance their knowledge.

Estimation Online Quiz with Answers

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Statistical inference is a branch of statistics in which we conclude (make wise decisions) about the population parameter by making use of sample information. To draw wise decisions, one can use estimation and hypothesis testing techniques based on extracted information from descriptive statistics. Statistical inference can be further divided into the Estimation of parameters and testing of the hypothesis.

Statistical estimation is the foundation of learning about a population by analyzing a sample. It’s essentially making educated guesses about population characteristics (parameters) based on the data we collect (samples).

Estimation Online Quiz

Estimation is a way of finding the unknown value of the population parameter from the sample information by using an estimator (a statistical formula) to estimate the parameter. One can estimate the population parameter by using two approaches (I) Point Estimation and (ii) Interval Estimation.

In point Estimation, a single numerical value is computed for each parameter, while in interval estimation a set of values (interval) for the parameter is constructed. The width of the confidence interval depends on the sample size and confidence coefficient. However, it can be decreased by increasing the sample size. The estimator is a formula used to estimate the population parameter by making use of sample information.

Various techniques for statistical estimation depends on the type of data and parameter of interest begin estimated. The following are a few techniques for statistical estimation:

  • Mean Estimation: Sample mean is used to estimate the population mean for continuous data.
  • Proportion Estimation: Sample proportion is used to estimate the population proportion for categorical data (e.g., yes/ no response).
  • Regression Analysis: Used to estimate relationships between variables and make predictions about a dependent variable based on an independent variable.
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Statistical estimation is a powerful tool that allows us to:

  • Move beyond the sample: Make generalizations about the population from which the data came.
  • Quantify uncertainty: Acknowledge the inherent variability in using samples and express the margin of error in the estimates.
  • Guide decision-making: Inform choices based on the best available information about the population.

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Online Estimation Quiz 7

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Online Estimation Quiz from Statistical Inference covers the topics of Estimation and Hypothesis Testing for the preparation of exams and different statistical job tests in Government/ Semi-Government or Private Organization sectors. These tests are also helpful in getting admission to different colleges and Universities. The online MCQS estimation quiz will help the learner understand the related concepts and enhance their knowledge.

MCQs about statistical inference covering the topics estimation, estimator, point estimate, interval estimate, properties of a good estimator, unbiasedness, efficiency, sufficiency, Large sample, and sample estimation.

1. If $Var(T_2)<Var(T_1)$ then $T_2$ is

 
 
 
 

2. A statistician calculates a 95% confidence interval for $\mu$ when $\sigma$ is known. The confidence interval is Rs 18000 to 22000, and then amount of sample means $\overline{X}$ is:

 
 
 
 

3. A confidence interval will be widened if:

 
 
 
 

4. Criteria to check a point estimator to be good are

 
 
 
 

5. The width of the confidence interval decreases if the confidence coefficient is

 
 
 
 

6. If the population Standard Deviation is unknown and the sample size is less than 30, then the Confidence Interval for the population mean ($\mu$) is

 
 
 
 

7. The best estimator of population proportion ($\pi$) is:

 
 
 
 

8. Which is NOT the property of a point estimator?

 
 
 
 

9. Interval estimation and confidence interval are:

 
 
 
 

10. In a $Z$-test the number of degrees of freedom is

 
 
 
 

11. In applying t-test

 
 
 
 

12. If $\mu=130, \overline{X}=150, \sigma=5$, and $n=10$. What Statistic is appropriate.

 
 
 
 

13. The consistency of an estimator can be checked by comparing

 
 
 
 

14. t-distribution is used when

 
 
 
 

15. For $\alpha=0.05$, the critical value of $Z_{0.05}$ is equal to

 
 
 
 

16. A sample is considered a small sample if the size is

 
 
 
 

17. For a biased estimator $\hat{\theta}$ of $\theta$, which one of the following is correct.

 
 
 
 

18. By decreasing $\overline{X}$ the length of the confidence interval for $\mu$

 
 
 
 

19. A large sample contains more than

 
 
 
 

20. If $1-\alpha=0.90$ then value of $Z_{\frac{\alpha}{2}}$ is

 
 
 
 

Statistical inference is a branch of statistics in which we conclude (make some wise decisions) about the population parameter using sample information. Statistical inference can be further divided into the Estimation of the Population Parameters and the Hypothesis Testing.

Estimation is a way of finding the unknown value of the population parameter from the sample information by using an estimator (a statistical formula) to estimate the parameter. One can estimate the population parameter by using two approaches (I) Point Estimation and (ii) Interval Estimation.

Online Estimation Quiz

  • A large sample contains more than
  • A sample is considered a small sample if the size is
  • In applying t-test
  • t-distribution is used when
  • If the population Standard Deviation is unknown and the sample size is less than 30, then the Confidence Interval for the population mean ($\mu$) is
  • If $\mu=130, \overline{X}=150, \sigma=5$, and $n=10$. What Statistic is appropriate?
  • If $1-\alpha=0.90$ then value of $Z_{\frac{\alpha}{2}}$ is
  • For $\alpha=0.05$, the critical value of $Z_{0.05}$ is equal to
  • In a $Z$-test the number of degrees of freedom is
  • The width of the confidence interval decreases if the confidence coefficient is
  • By decreasing $\overline{X}$ the length of the confidence interval for $\mu$
  • A statistician calculates a 95% confidence interval for $\mu$ when $\sigma$ is known. The confidence interval is Rs 18000 to 22000, and then the amount of sample means $\overline{X}$ is:
  • Criteria to check a point estimator to be good are
  • The consistency of an estimator can be checked by comparing
  • If $Var(T_2)<Var(T_1)$ then $T_2$ is
  • For a biased estimator $\hat{\theta}$ of $\theta$, which one of the following is correct?
  • Which is NOT the property of a point estimator?
  • The best estimator of population proportion ($\pi$) is:
  • Interval estimation and confidence interval are:
  • A confidence interval will be widened if:

In point estimation, a single numerical value is computed for each parameter, while in an interval estimation, a set of values (interval) for the parameter is constructed. The width of the confidence interval depends on the sample size and confidence coefficient. However, it can be decreased by increasing the sample size. The estimator is a formula used to estimate the population parameter by making use of sample information.

Online Estimation Quiz

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