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.

Estimation Statistics MCQs 3

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Estimation Statistics MCQs Quiz covers the topics of Estimate and Estimation 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 Estimation Statistics MCQs Quiz will help the learner to understand the related concepts and enhance their knowledge too.

This MCQs quiz is about statistical inference. It will help you to understand the basic concepts related to Inferential statistics. This test will also help you to prepare yourself for different exam related to education or jobs.

1. A range (set) of values within which the population parameter is expected to occur is called:

 
 
 
 

2. A single value used to estimate a population value is called:

 
 
 
 

3. If $\hat{\theta}$ is the estimator of the parameter $\theta$, then $\hat{\theta}$ is called unbiased if:

 
 
 
 

4. The estimator is said to be ___________ if the mean of the estimator is not equal to the mean of the population parameter.

 
 
 
 

5. The value of a statistic tends towards the value of the population as the sample size increases. What is it said to be?

 
 
 
 

6. The process of making estimates about the population parameter from a sample is called

 
 
 
 

7. The estimate is the observed value of an

 
 
 
 

8. ‘Statistic’ is an estimator and its computer values are called:

 
 
 
 

9. There are two main branches of statistical inference, namely

 
 
 
 

10. The numerical value which we determine from the sample for a population parameter is called

 
 
 
 

11. The process of using sample data to estimate the values of unknown population parameters is called

 
 
 
 

12. For computing the confidence interval about a single population variance, the following test will be used

 
 
 
 

13. Estimation can be classified into

 
 
 
 

14. The difference between the two end points of a confidence interval is called

 
 
 
 

15. The end points of a confidence interval are called

 
 
 
 

16. The estimate is the observed value of an:

 
 
 
 

17. The end points of a confidence interval are called:

 
 
 
 

18. A set (range) of values calculated from the sample data and is likely to contain the true value of the parameter with some probability is called:

 
 
 
 

19. A formula or rule used for estimating the parameter of interest is called:

 
 
 
 

20. The probability associated with confidence interval is called

 
 
 
 

Statistical inference is a branch of statistics in which we conclude (make wise decisions) about the population parameter by making use of sample information. Statistical inference can be further divided into the Estimation of parameters and testing of the hypothesis.

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.

Estimation, point estimate and Interval Estimate

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.

Estimation Statistics

Online Estimation Statistics MCQs

  • The process of making estimates about the population parameter from a sample is called
  • There are two main branches of statistical inference, namely
  • Estimation can be classified into
  • A formula or rule used for estimating the parameter of interest is called:
  • ‘Statistic’ is an estimator and its computer values are called:
  • The estimate is the observed value of an:
  • The process of using sample data to estimate the values of unknown population parameters is called
  • The numerical value which we determine from the sample for a population parameter is called
  • A single value used to estimate a population value is called:
  • A set (range) of values calculated from the sample data and is likely to contain the true value of the parameter with some probability is called:
  • A range (set) of values within which the population parameter is expected to occur is called:
  • The end points of a confidence interval are called:
  • The probability associated with confidence interval is called
  • The estimator is said to be ________ if the mean of the estimator is not equal to the mean of the population parameter.
  • If $\hat{\theta}$ is the estimator of the parameter $\theta$, then $\hat{\theta}$ is called unbiased if:
  • The value of a statistic tends towards the value of the population as the sample size increases. What is it said to be?
  • For computing the confidence interval about a single population variance, the following test will be used
  • The end points of a confidence interval are called
  • The difference between the two end points of a confidence interval is called
  • The estimate is the observed value of an
Estimation Statistics MCQs

Estimation is a fundamental part of statistics because populations can be very large or even infinite, making it impossible to measure every single member. By using estimation techniques, we can draw conclusions about the bigger picture from a manageable amount of data.

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The Z-Score Definition, Formula, Real Life Examples (2020)

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Z-Score Definition: The Z-Score also referred to as standardized raw scores (or simply standard score) is a useful statistic because not only permits to computation of the probability (chances or likelihood) of the raw score (occurring within normal distribution) but also helps to compare two raw scores from different normal distributions. The Z score is a dimensionless measure since it is derived by subtracting the population mean from an individual raw score and then this difference is divided by the population standard deviation. This computational procedure is called standardizing raw score, which is often used in the Z-test of testing of hypothesis.

Any raw score can be converted to a Z-score formula by

$$Z-Score=\frac{raw score – mean}{\sigma}$$

Z-Score Real Life Examples

Example 1: If the mean = 100 and standard deviation = 10, what would be the Z-score of the following raw score

Raw ScoreZ Scores
90$ \frac{90-100}{10}=-1$
110$ \frac{110-100}{10}=1$
70$ \frac{70-100}{10}=-3$
100$ \frac{100-100}{10}=0$

Note that: If Z-Score,

  • has a zero value then it means that the raw score is equal to the population mean.
  • has a positive value then it means that the raw score is above the population mean.
  • has a negative value then it means that the raw score is below the population mean.
The Z-Score Definition, Formula, Real Life Examples

Example 2: Suppose you got 80 marks in an Exam of a class and 70 marks in another exam of that class. You are interested in finding that in which exam you have performed better. Also, suppose that the mean and standard deviation of exam-1 are 90 and 10 and in exam-2 60 and 5 respectively. Converting both exam marks (raw scores) into the standard score, we get

$Z_1=\frac{80-90}{10} = -1$

The Z-score results ($Z_1=-1$) show that 80 marks are one standard deviation below the class mean.

$Z_2=\frac{70-60}{5}=2$

The Z-score results ($Z_2=2$) show that 70 marks are two standard deviations above the mean.

From $Z_1$ and $Z_2$ means that in the second exam, students performed well as compared to the first exam. Another way to interpret the Z score of $-1$ is that about 34.13% of the students got marks below the class average. Similarly, the Z Score of 2 implies that 47.42% of the students got marks above the class average.

Application of Z Score

  • Identifying Outliers: The standard score can help in identifying the outliers in a dataset. By looking for data points with very high negative or positive z-scores, one can easily flag potential outliers that might warrant further investigation.
  • Comparing Data Points from Different Datasets: Z-scores allow us to compare data points from different datasets because these scores are expressed in standard deviation units.
  • Standardizing Data for Statistical Tests: Some statistical tests require normally distributed data. The Zscore can be used to standardize data (transforming it to have a mean of 0 and a standard deviation of 1), making it suitable for such tests.

Limitation of ZScores

  • Assumes Normality: The Zscores are most interpretable when the data is normally distributed (a bell-shaped curve). If the data is significantly skewed, the scores might be less informative.
  • Sensitive to Outliers: The presence of extreme outliers can significantly impact the calculation of the mean and standard deviation, which in turn, affects the standard score of all data points.

In conclusion, z-scores are a valuable tool for understanding the relative position of a data point within its dataset. The standard score offers a standardized way to compare data points, identify outliers, and prepare data for statistical analysis. However, it is important to consider the assumptions of normality and the potential influence of outliers when interpreting the z-scores.

Read about Standard Normal Table

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Sufficient Estimators and Sufficient Statistics

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Introduction to Sufficient Estimator and Sufficient Statistics

An estimator $\hat{\theta}$ is sufficient if it makes so much use of the information in the sample that no other estimator could extract from the sample, additional information about the population parameter being estimated.

The sample mean $\overline{X}$ utilizes all the values included in the sample so it is a sufficient estimator of the population mean $\mu$.

Sufficient estimators are often used to develop the estimator that has minimum variance among all unbiased estimators (MVUE).

If a sufficient estimator exists, no other estimator from the sample can provide additional information about the population being estimated.

If there is a sufficient estimator, then there is no need to consider any of the non-sufficient estimators. A good estimator is a function of sufficient statistics.

Let $X_1, X_2,\cdots, X_n$ be a random sample from a probability distribution with unknown parameter $\theta$, then this statistic (estimator) $U=g(X_1, X_,\cdots, X_n)$ observation gives $U=g(X_1, X_2,\cdots, X_n)$ does not depend upon population parameter $\Theta$.

Sufficient Statistics Example

The sample mean $\overline{X}$ is sufficient for the population mean $\mu$ of a normal distribution with known variance. Once the sample mean is known, no further information about the population mean $\mu$ can be obtained from the sample itself, while the median is not sufficient for the mean; even if the median of the sample is known, knowing the sample itself would provide further information about the population mean $\mu$.

Mathematical Definition of Sufficiency

Suppose that $X_1,X_2,\cdots,X_n \sim p(x;\theta)$. $T$ is sufficient for $\theta$ if the conditional distribution of $X_1,X_2,\cdots, X_n|T$ does not depend upon $\theta$. Thus
\[p(x_1,x_2,\cdots,x_n|t;\theta)=p(x_1,x_2,\cdots,x_n|t)\]
This means that we can replace $X_1,X_2,\cdots,X_n$ with $T(X_1,X_2,\cdots,X_n)$ without losing information.

Sufficient Estimator Sufficient Statistics

For further reading visit: https://en.wikipedia.org/wiki/Sufficient_statistic

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