Unbiasedness

Unbiasedness is a statistical concept that describes the accuracy of an estimator. An estimator is said to be an unbiased estimator if its expected value (or average value over many samples) equals the corresponding population parameter, that is, $E(\hat{\theta}) = \theta$.

If the expected value of an estimator $\theta$ is not equal to the corresponding parameter then the estimator will be biased. The bias of an estimator of $\hat{\theta}$ can be defined as

$$Bias = E(\hat{\theta}) – \theta$$

Note that $\overline{X}$ is an unbiased estimator of the mean of a population. Therefore,

  • $\overline{X}$ is an unbiased estimator of the parameter $\mu$ in Normal distribution.
  • $\overline{X}$ is an unbiased estimator of the parameter $p$ in the Bernoulli distribution.
  • $\overline{X}$ is an unbiased estimator of the parameter $\lambda$ in the Poisson distribution.
Unbiasedness, positive bias, negative bias, unbiased

However, the expected value of the sample variance $S^2=\frac{\sum\limits_{i=1}^n (X_i – \overline{X})^2 }{n}$ is not equal to the population variance, that is $E(S^2) = \sigma^2$.

Therefore, sample variance is not an unbiased estimator of the population variance $\sigma^2$.

Note that it is possible to have more than one unbiased estimator for an unknown parameter. For example, the sample mean and sample median are both unbiased estimators of the population mean $\mu$ if the population distribution is symmetrical.

Question: Show that the sample mean is an unbiased estimator of the population mean.

Solution:

Let $X_1, X_2, \cdots, X_n$ be a random sample of size $n$ from a population having mean $\mu$. The sample mean is $\overline{X}$ is

$$\overline{X} = \frac{1}{n} \sum\limits_{i=1}^n X_i$$

We must show that $E(\overline{X})=\mu$, therefore, taking the expectation on both sides,

\begin{align*}
E(\overline{X}) &= E\left[\frac{1}{n} \Sigma X_i \right]\\
&= \frac{1}{n} E(X_i) = \frac{1}{n} E(X_1 + X_2 + \cdots + X_n)\\
&= \frac{1}{n} \left[E(X_1) + E(X_2) + \cdots + E(X_n) \right]
\end{align*}

Since, in the random sample, the random variables $X_1, X_2, \cdots, X_n$ are all independent and each has the same distribution of the population, then $E(X_1)=E(X_2)=\cdots=E(X_n)$. So,

$$E(\overline{x}) = \frac{1}{n}(\mu+\mu+\cdots + \mu) = \mu$$

Why Unbiasedness is Important

  • Accuracy: Unbiasedness is a measure of accuracy, not precision. Unbiased estimators provide accurate estimates on average, reducing the risk of systematic errors. However, an unbiased estimator can still have a large variance, meaning its individual estimates can be far from the true value.
  • Consistency: An unbiased estimator is not necessarily consistent. Consistency refers to the tendency of an estimator to converge to the true value as the sample size increases.
  • Foundation for Further Analysis: Unbiased estimators are often used as building blocks for more complex statistical procedures.

Unbiasedness Example

Imagine you’re trying to estimate the average height of students in your university. If you randomly sample 100 students and calculate their average height, this average is an estimator of the true average height of all students in that university. If this average height is consistently equal to the true average height of the entire student population, then your estimator is unbiased.

Unbiasedness is the state of being free from bias, prejudice, or favoritism. It can also mean being able to judge fairly without being influenced by one’s own opinions. In statistics, it also refers to (i) A sample that is not affected by extraneous factors or selectivity (ii) An estimator that has an expected value that is equal to the parameter being estimated.

Applications and Uses of Unbiasedness

  • Parameter Estimation:
    • Mean: The sample mean is an unbiased estimator of the population mean.
    • Variance: The sample variance, with a slight adjustment (Bessel’s correction), is an unbiased estimator of the population variance.
    • Regression Coefficients: In linear regression, the ordinary least squares (OLS) estimators of the regression coefficients are unbiased under certain assumptions.
  • Hypothesis Testing:
    • Unbiased estimators are often used in hypothesis tests to make inferences about population parameters. For example, the t-test for comparing means relies on the assumption that the sample means are unbiased estimators of the population means.
  • Machine Learning: In some machine learning algorithms, unbiased estimators are preferred for model parameters to avoid systematic errors.
  • Survey Sampling: Unbiased sampling techniques, such as simple random sampling, are used to ensure that the sample is representative of the population and that the estimates obtained from the sample are unbiased.

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MCQs Demography Quiz 1

The post is about MCQs Demography Quiz with Answers. All these MCQs related to demography (population studies) will also help you understand the concepts related to people for the preparation of different examinations. Test your knowledge and learn something new about the fascinating world of demography! Let us start with the MCQs demography quiz now.

Online Multiple-Choice Questions about Demography Quiz with Answers

1. Female crude death rate generally ———– male crude death rate.

 
 
 
 

2. Death rate computed for a particular specified section of the population is termed as ———-.

 
 
 
 

3. ——— is defined as the number of deaths per 1000 population in a specific community or region.

 
 
 
 

4. Which of the following is a mortality indicator?

 
 
 
 

5. A mortality indicator among the following is

 
 
 
 

6. What is the maternal mortality rate?

 
 
 
 

7. In a pyramid representation, male bars are usually on the ————.

 
 
 
 

8. The death rate specific to age and sex overcomes the drawback of ———.

 
 
 
 

9. The most universally accepted indicator of health status is?

 
 
 
 

10. The ———— is defined as the chance of dying of a newly born infant within a year of its life, under the given mortality conditions.

 
 
 
 

11. Which of the following can be used to access the health status of a country?

 
 
 
 

12. Which of the following cannot be drawn from the population pyramid of a country?

 
 
 
 

13. Which country has the maximum life expectancy?

 
 
 
 

14. Which among the following is not a measure of mortality?

 
 
 
 

15. ———– is calculated as a weighted average of the age-specific death rate of a given population.

 
 
 
 

16. Which of the following rates completely ignores the age and sex distribution of the population?

 
 
 
 

17. The formula for calculating the crude death rate is

 
 
 
 

18. If birth rate > death rate, then there is a/an ————- trend.

 
 
 
 

19. The death rate is also called ———–

 
 
 
 

20. Given the zero growth of the life table, the population is also known as a ———– population.

 
 
 
 

Online MCQs Demography Quiz

  • The death rate is also called ———–
  • Which among the following is not a measure of mortality?
  • ——— is defined as the number of deaths per 1000 population in a specific community or region.
  • The formula for calculating the crude death rate is
  • Which of the following rates completely ignores the age and sex distribution of the population?
  • Female crude death rate generally ———– male crude death rate.
  • The death rate specific to age and sex overcomes the drawback of ———.
  • Death rate computed for a particular specified section of the population is termed as ———-.
  • ———– is calculated as a weighted average of the age-specific death rate of a given population.
  • If birth rate > death rate, then there is a/an ————- trend.
  • The ———— is defined as the chance of dying of a newly born infant within a year of its life, under the given mortality conditions.
  • What is the maternal mortality rate?
  • Given the zero growth of the life table, the population is also known as a ———– population.
  • In a pyramid representation, male bars are usually on the ————.
  • Which of the following is a mortality indicator?
  • A mortality indicator among the following is
  • Which of the following can be used to access the health status of a country?
  • The most universally accepted indicator of health status is?
  • Which country has the maximum life expectancy?
  • Which of the following cannot be drawn from the population pyramid of a country?
MCQs Demography quiz with Answers

Keywords: demography quiz, population quiz, demographics test, population studies, population growth, migration, age structure, fertility, mortality, population distribution, online quiz, learn demography

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Online Demography Quizzes

These Demography Quizzes will challenge your understanding of key demographic concepts, including population growth, migration, age structure, etc. Acquire Demography concepts from multiple-choice type questions. All these MCQs related to demography quizzes will also help you understand the concepts related to people for the preparation of different examinations. Test your knowledge and learn something new about the fascinating world of demography!

Demography Quiz 3
Demography Quiz 2Demography Quiz 1

Demography Quizzes Instructions

  • Read Carefully: Each question will present multiple-choice options. Carefully read each question and all the available options before selecting your response.
  • Choose Wisely: Select the answer you believe is most accurate and finally click “Submit” to get the grade on the quiz.
  • Review Your Answers: After submitting your answers, you will receive a score and the correct answers.
  • Learn & Improve: Use this quiz as an opportunity to learn more about demography. Review the questions you missed and explore the topics in greater depth.
Online Demography Quizzes

Keywords: demography quiz, population quiz, demographics test, population studies, population growth, migration, age structure, fertility, mortality, population distribution, online quiz, learn demography

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Solved Binomial Distribution Questions

This post is about some solved Binomial distribution Questions. These solved binomial distribution questions make use of computation of (i) the exact probability case, (ii) at least case, (iii) at most case, and (iv) between cases.

Binomial-Probability-Distribution
Binomial distribution questions
  • The sum of all probabilities in the distribution sums up to 1
  • The probability of success in all $n$ trials is $p^n$
  • The probability of failure in all $n$ trials is $(1 – p)^n = q^n$
  • Probability of success in at least one trial = $P(X \ge 1) = 1 – P(X = 0) = 1 – q^n$.
  • Probability of at least $x$ successes = $P(X \ge x) = \sum\limits_{x} \binom{n}{x}p^xq^{n-x}\quad (x = x, x + 1,\cdots, n$)
  • Probability of at most $x$ successes = $P(X \le x) =\sum\limits_{x} \binom{n}{x}p^x q^{n-x}\quad (x=0,1,\cdots,x)$
  • If in $n$ trials, the experiment is repeated $N$ times, the expected frequencies are $N\cdot P(x)$ for $x = 0, 1, 2, 3, \cdots, n$.

Solved Binomial Distribution Questions

Question 1: A die is rolled 5 times and a 5 or 6 is considered a success. Find the probability of (i) no success, (ii) at least 2 successes, (iii) at least one but not more than 3 successes.

Solution:

The Sample Space is $S=\{1, 2, 3, 4, 5, 6\}$. Since the occurrence of 5 or 6 is considered a success, therefore, $p=\frac{2}{6}=\frac{1}{3} \Rightarrow q=1-p = 1-\frac{1}{3} = \frac{2}{3}$.

(i) No success

$n=5, p=\frac{1}{3}, q=\frac{2}{3}$, x=0$

\begin{align*}
P(X=x) &= \binom{n}{x}p^x q^{n-x}\\
P(X=0) &= \binom{5}{0} \left(\frac{1}{3}\right)^0\left(\frac{2}{3}\right)^5=0.1316
\end{align*}

(ii) At least 2 successes

\begin{align*}
P(X \ge 2) & = 1 – P(X<2)\\
&= 1 – [P(X=0) + P(X=1)]\\
&= 1- [0.13168 + 0.3292] = 0.5391
\end{align*}

(iii) At least one but not more than 3 successes

\begin{align*}
P(1 \le x \le 3) &= P(X=1) + P(X=2) + P(X=3)\\
&= 0.3292 + 0.3292 + 0.1646 = 0.823
\end{align*}

Question 2: Find the probability of getting (i) exactly 4 heads and (ii) not more than 4 heads when 6 coins are tossed.

Solution:

From the given information, $n = 6, x = 4, p = q = \frac{1}{2}$

(i) Exactly 4 heads

\begin{align*}
P(X=x) &= \binom{n}{x} p^x q^{n-x}\\
&= \binom{6}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{6-4} = 0.234
\end{align*}

(ii) Not more than 4 heads

\begin{align*}
P(X\le 4) & = 1 – p(X\ge 4)\\
&= 1 – P(X=4) + P(X=5) + P(X=6)
\end{align*}

Question 3: If 60% of the voters in a large district prefer candidate-A, what is the probability that in a sample of 12 voters, exactly 7 will prefer A?

Solution:

From given information in the questions, $p=06, q=0.4, n=12, x=7$

\begin{align*}
P(X=x)&= \binom{n}{x}p^x q^{n-x}\\
P(X=7) &= \binom{12}{7} (0.6)^7(0.4)^5&= 0.227
\end{align*}

Question 4: The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?

Solution:

From the given information in the question, $n=7, x=5, p=0.9, q=0.10$

\begin{align*}
P(X=x)&= \binom{n}{x}p^x q^{n-x}\\
P(X=5) &= \binom{7}{5}(0.9)^5(0.1)^2 = 0.124
\end{align*}

Question 5: The incidence of occupational disease in an industry is such that the workmen have a 20% chance of suffering from it. What is the probability that out of 6 workmen (i) not more than 2, and (ii) 4 or more will catch the disease?

Solution:

From the given information in the questions

Probability of suffering from occupational disease = $\frac{20}{100}=\frac{1}{5}=0.20$

Probability of not suffering from occupational disease = $1 – \frac{1}{5} = \frac{4}{5}=0.80$

(i) Probability that out of 6 workers, not more than two will suffer

\begin{align*}
P(X\le 2) &= \binom{6}{0}\left(\frac{4}{5}\right)^0 \left(\frac{1}{5}\right)^6 + \binom{6}{1}\left(\frac{4}{5}\right)^1 \left(\frac{1}{5}\right)^5 + \binom{6}{2}\left(\frac{4}{5}\right)^2 \left(\frac{1}{5}\right)^4\\
&=0.01696
\end{align*}

(ii) Probability that out of 6 workers, 4 or more will suffer

\begin{align*}
P(X\ge 4) &= \binom{6}{4}\left(\frac{4}{5}\right)^4 \left(\frac{1}{5}\right)^2 + \binom{6}{5}\left(\frac{4}{5}\right)^5 \left(\frac{1}{5}\right)^1 + \binom{6}{6}\left(\frac{4}{5}\right)^6 \left(\frac{1}{5}\right)^0\\
&=0.90112
\end{align*}

Question 6: A multiple-choice has 15 questions, each with 4 possible answers of which only 1 is the correct answer. What is the probability that sheer guesswork yields from 5 to 10 correct answers?

Solution:

Probability of answering any question correctly: $p=\frac{1}{4}=0.25$

Probability of answering any question wrongly: $q=\frac{3}{4}=0.75$

\begin{align*}
P(5 \le x \le 10) &= P(X=5) + P(X=6) + \cdots + P(X=10)\\
&=\binom{15}{5}\left(\frac{1}{4}\right)^5\left(\frac{3}{4}\right)^{10}+\cdots + \binom{15}{10}\left(\frac{1}{4}\right)^5\left(\frac{3}{4}\right)^{5} \\
&= 0.31339
\end{align*}

Question 7: A commuter drivers to work each morning. The route she takes each day includes ten stoplights. Assume the probability each stoplight is red when she gets to it is 0.2 and that these stoplights (trials) are independent. What is the distribution of $X$, the number of times she must stop for a red light on her way to work? Evaluates $P(X=0) and $P(X<3).

Solution:

The distribution of $X$ is binomial because trials are independent. The probability of getting red spotlight (success) is 0.2 which remains the same, the number of trials is fixed ($n=10$).

The further information given in the Question is: $n=10, p=0.2, q=0.8$

\begin{align*}
P(X=0)&=\binom{10}{0} (0.2)^0(0.8)^{10} = 0.10737\\
P(X<3) &=\binom{10}{0} (0.2)^0(0.8)^{10} + \binom{10}{1} (0.2)^1(0.8)^{9} + \binom{10}{2} (0.2)^2(0.8)^{8} = 0.6777
\end{align*}

Application of Binomial Probability Distribution

  • Quality Control:
    • Assessing Product Reliability: Manufacturers use binomial distribution to estimate the probability of defective products in a batch, helping them maintain quality standards.
    • Predicting Failure Rates: By analyzing past data, companies predict the likelihood of equipment failure using a binomial probability distribution, aiding in preventive maintenance and reducing downtime.
  • Genetics:
    • Predicting Inheritance Patterns: In genetics, Binomial distribution helps to predict the probability of offspring inheriting specific traits based on parental genotypes.
    • Analyzing Genetic Mutations: Binomial distribution is used to study the frequency of genetic mutations in populations.
  • Medicine:
    • Clinical Trials: Binomial distribution is essential for designing and analyzing clinical trials, assessing the effectiveness of treatments, and determining the probability of side effects.
    • Epidemiology: Binomial distribution helps to model the spread of infectious diseases and predict outbreak risks.
  • Finance:
    • Risk Assessment: Financial institutions use Binomial Probability Distribution to assess the risk of loan defaults or investment failures.
    • Option Pricing: Binomial probability distribution is a key component of option pricing models, helping to determine the fair value of options contracts.
  • Social Sciences:
    • Survey Analysis: Binomial distribution is used to analyze survey data, such as predicting voter behavior or public opinion on specific issues.
    • Market Research: Binomial Probability Distribution helps businesses to understand consumer preferences and predict market trends.

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