Estimates and Estimation

The post is about Estimation MCQs with Answers. MCQs are all about statistical inference and cover the topics of estimation, estimator, point estimate, interval estimate, properties of a good estimator, unbiasedness, efficiency, sufficiency, Large sample, and sample estimation. There are 20 multiple-choice questions from the estimation section. Let us start with Estimation MCQs with Answers.

Estimation MCQs with Answers

• If $E(\hat{\theta})=\theta$ then $\hat{\theta}$ is called
• A statistic $\hat{\theta}$ is said to be an unbiased estimator of $\theta$, if
• The following statistics are unbiased
• The following is an unbiased estimator of the population variance $\sigma^2$
• In point estimation we get
• The formula used to estimate a parameter is called
• A specific value calculated from a sample is called
• A function that is used to estimate a parameter is called
• $1-\alpha$ is called
• The level of confidence is denoted by
• The other name of the significance level is
• What will be the confidence level if the level of significance is 5% (0.05)
• The probability that the confidence interval does not contain the population parameter is denoted by
• The probability that the confidence interval does contain the parameter is denoted by
• The way of finding the unknown value of the population parameter from the sample values by using a formula is called ——–.
• There are four steps involved with constructing a confidence interval. What is typically the first one?
• What happens as a sample size gets larger?
• After identifying a sample statistic, what is the proper order of the next three steps of constructing a confidence interval?
• Testing of hypothesis may be replaced by?
• A point estimate is often insufficient. Why?

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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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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
• Testing of Hypothesis (Hypothesis Testing)

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.

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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