## Estimation of Population Parameters

### Introduction to Estimation of Population Parameters

In statistics, estimating population parameters is important because it allows the researcher to conclude a population (whole group) by analyzing a small part of that population. The estimation of population parameters is done when the population under study is large enough. For example, instead of performing a census, a random sample from the population can be drawn. To draw some conclusions about the population, one can calculate the required sample statistic(s).

### Important Terminologies

The following are some important terminologies to understand the concept of estimating the population parameters.

• Population: The entire collection of individuals or items one is interested in studying. For instance, all the people living in a particular country.
• Sample: A subgroup (or small portion) chosen from the population that represents the larger group.
• Parameter: A characteristic that describes the entire population, such as the population mean, median, or standard deviation.
• Statistic: A value calculated from the sample data used to estimate the population parameter. For example, the sample mean is an estimate of the population mean. It is the characteristics of the sample under study.

Various statistical methods are used to estimate population parameters with different levels of accuracy. The accuracy of the estimate depends on the size of the sample and how well the sample represents the population.

We use statistics calculated from the sample data as estimates for the population parameters.

• Sample mean: is used to estimate the population mean. It is calculated by averaging the values of all observations in the sample, that is the sum of all data values divided by the total number of observations in the data.
• Sample proportion: is used to estimate the population proportion (percentage). It represents the number of successes (events of interest) divided by the total sample size.
• Sample standard deviation: is used to estimate the population standard deviation. It reflects how spread out the data points are in the sample.

### Types of Estimates

There are two types of estimates:

• Point Estimate: A single value used to estimate the population parameter. The example of point estimates are:
• The mean/average height of Boys in Colleges is 65 inches.
• 65% of Lahore residents support a ban on cell phone use while driving.
• Interval Estimate: It is a set of values (interval) that is supposed to contain the population parameter. Examples of interval estimates are:
• The mean height of Boys in Colleges lies between 63.5 and 66.5 inches.
• 65% ($\pm 3$% of Lahore residents support a ban on cell phone use during driving.

### Some Examples

Estimation of population parameters is widely used in various fields of life. For example,

• a company might estimate customer satisfaction through a sample survey,
• a biologist might estimate the average wingspan of a specific bird species by capturing and measuring a small group.

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## Statistical Inference: An Introduction

### 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:

### 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.

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## Best Estimation Online Quiz

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. These Estimation online quiz will also be helpful in getting 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

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 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.

There are various techniques for statistical estimation that depends on the type of data and parameter of interest begin estimated. The followings are 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.

The 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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MCQs General Knowledge

## Important Online Estimation Quiz 7

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. Criteria to check a point estimator to be good are

2. Interval estimation and confidence interval are:

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

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

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

6. In applying t-test

7. A large sample contains more than

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

9. A confidence interval will be widened if:

10. 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:

11. 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

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

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

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

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

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

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

18. t-distribution is used when

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

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

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