Unbiased Estimators and Their Efficiency

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Questions and Answers

What defines the most efficient estimator among unbiased estimators?

It is always the sample median.

It has the largest sample size.

It depends on the choice of biased estimators.

It has the smallest variance. (correct)

How is relative efficiency of two unbiased estimators defined?

It is the ratio of their variances. (correct)

It is the average of their variances.

It is the difference of their variances.

It is the sum of their variances.

Which unbiased estimator is more efficient based on the given variances: $Var(x̄) = \sigma^2/n$ and $Var(x.5) = \frac{\pi^2 \sigma^2}{n}$?

Neither can be determined.

Both have equal efficiency.

Sample mean is more efficient. (correct)

Sample median is more efficient.

If $Var(\hat{\theta_1}) < Var(\hat{\theta_2})$, how does this affect their relative efficiency?

The relative efficiency is greater than 1. Signup and view all the answers

Which condition is necessary for the sample mean to be considered more efficient than the sample median?

The variance of the sample mean must be less. Signup and view all the answers

What does the symbol $µN$ represent in the given definitions?

The average of $xNi$ values Signup and view all the answers

What is the role of $SN$ in the context provided?

It measures the spread of the $xNi$ values Signup and view all the answers

In the limit as $N$ tends towards infinity, what behavior does $mN$ demonstrate?

It approaches the maximum value of $xNi$ Signup and view all the answers

Condition (4) in the definitions implies which of the following conditions must hold?

$n$ and $N$ must both tend towards infinity Signup and view all the answers

What expression is defined for $x̄$ in the context?

$x̄ = ∑_{i=1}^{n} R_i xNi$ Signup and view all the answers

The notation $mN = ext{max}(xNi - µN)²$ represents what?

The largest squared deviation from the mean Signup and view all the answers

What implication does the expression $Nn =: α²[0, 1]$ have?

Indicates a relationship for variance scaling Signup and view all the answers

What transformation is applied to $xNi$ in the scaling process?

$xNi / SN$ Signup and view all the answers

What happens to the sampling distribution of the sample mean as the sample size increases?

It approaches a normal distribution. Signup and view all the answers

What is the variance of the sampling distribution of x̄ when the sample size is n?

$\frac{\sigma^2}{n}$ Signup and view all the answers

In an example where the population mean is µ = 8 and σ = 3 with a sample size of 36, what is the standard deviation of the sampling distribution?

$\frac{3}{\sqrt{36}}$ Signup and view all the answers

What is the probability that the sample mean is between 7.8 and 8.2 for the given parameters?

0.3108 Signup and view all the answers

Who contributed significantly to the history of the Central Limit Theorem?

Jarl W. Lindeberg and Paul P. Lévy Signup and view all the answers

If a population is not normally distributed, under what condition can the Central Limit Theorem still be applied?

n > 25 Signup and view all the answers

What does CLT stand for in the context of sampling distributions?

Central Limit Theorem Signup and view all the answers

What is the purpose of the Central Limit Theorem in statistics?

To show that the sampling distribution of the sample mean approaches a normal distribution. Signup and view all the answers

What does the condition $rac{mN}{n} eq 0$ signify when $\alpha = 0$?

It shows that the sample size is effective. Signup and view all the answers

Which of the following is true when $eta eq 1$?

The ratio $N - n o rac{1}{1 - eta}$. Signup and view all the answers

What happens to $S_N$ as $N o ext{infinity}$ under condition (4)?

$S_N$ converges to a positive number. Signup and view all the answers

In the context of sampling distributions, what does $X hickapprox ext{Binomial}(n, p)$ represent?

The sum of multiple Bernoulli trials. Signup and view all the answers

What is the formula for the variance of the sample proportion $ar{p}$ derived from a Bernoulli distribution?

$rac{p(1-p)}{n}$ Signup and view all the answers

What does $E[ar{p}] = p$ indicate about the estimator $ar{p}$?

It is an unbiased estimator of population proportion. Signup and view all the answers

What is the implication of $ ext{Var}(X_i) = p(1-p)$ for a Bernoulli distributed variable?

The variance is independent of sample size. Signup and view all the answers

What does $ar{x} o rac{ar{p}}{n}$ imply as $n$ increases?

The average becomes more accurate. Signup and view all the answers

What is the expected value of the sample variance $E[s^2]$ if the population variance is $σ^2$ and the sample size is $n$?

$σ^2$ Signup and view all the answers

Why do we lose one degree of freedom when calculating sample variance?

Because of the extra term $n(\bar{x} - \mu)^2$. Signup and view all the answers

If the standard deviation of the freezers is specified as no more than 4 degrees, what is the population variance?

16 Signup and view all the answers

What distribution does the sum of squared z-scores follow with $n$ degrees of freedom?

$χ^2$ Distribution Signup and view all the answers

In the equation $E[\sum_{i=1}^{n}(x_i - \bar{x})^2]$, what does the term $(x_i - \bar{x})$ represent?

The deviation of each sample from the sample mean. Signup and view all the answers

What does the notation $E[\sum_{i=1}^{n}(x_i - \mu)^2]$ indicate in the context of expected values?

Expected sum of population deviations. Signup and view all the answers

How do you compute the total sum of squares $\sum_{i=1}^{n}(x_i - \mu)^2$ in a sampling distribution?

By subtracting the population mean from each sample and squaring it. Signup and view all the answers

What correction is made in the computation of the sample variance when the population mean is unknown?

Use the sample mean instead of the population mean. Signup and view all the answers

What is the value of K for the sample variance if the population standard deviation is 4 and the probability of exceeding this limit is less than 0.05?

27.25 Signup and view all the answers

If the sample variance, s², is greater than which value, would there be strong evidence to suggest that the population variance exceeds 16?

27.52 Signup and view all the answers

What type of sampling does the unbiased estimator for Var(x̄) = s²/n pertain to?

Random sampling with replacement Signup and view all the answers

In random sampling without replacement, what is the expected value of the sample variance, E(s²)?

N/(N-1)σ² Signup and view all the answers

How is the unbiased estimator for Var(x̄) impacted in sampling without replacement compared to with replacement?

It factors in the population size to reduce variance. Signup and view all the answers

What is the critical value of χ² used to find K when n = 14?

22.36 Signup and view all the answers

If the population standard deviation increases, what is the effect on the upper limit K for the sample variance if maintaining a probability of exceeding this limit less than 0.05?

K increases. Signup and view all the answers

Which of the following expressions represents the unbiased estimator of variance for sampling without replacement?

s² N/(N - n) Signup and view all the answers

Study Notes

Lecture 5: Sampling Distribution Theory (Chapter 6)

The lecture covers sampling distribution theory, focusing on sampling from a population, sampling distributions of sample means, sample proportions, and sample variances, along with properties of point estimators.

Plan of This Lecture

Sampling from a Population
Sampling Distributions of Sample Means
Sampling Distributions of Sample Proportions
Sampling Distributions of Sample Variances
Properties of Point Estimators (Section 7.1)

Review of Descriptive and Inferential Statistics

Descriptive statistics involves collecting, presenting, and describing data.
Inferential statistics involves drawing conclusions and making decisions about a population based on sample data.
Estimation is used to estimate population parameters using sample data. Examples include estimating the population mean weight.
Hypothesis testing uses sample evidence to test claims about population parameters. An example is testing if the population mean weight is 120 pounds.

Sampling from a Population

Statistical analysis requires a proper sample from a population.
A population includes all items of interest.
A simple random sample involves randomly choosing n objects from a population, with each object having an equal chance of being selected.
Random sampling with replacement involves drawing an item from the population and placing it back before the next draw.
Random sampling without replacement involves drawing an item, not returning it to the population, and then drawing the next item.

Population and Simple Random Sample

Statistical analysis requires a sample representative of a population.
A population comprises all items of interest.
A large population can be treated as infinite for sampling purposes.
Random processes may underpin population generation.

Sampling Distributions

Random samples' randomness stems from random drawing and sampling without knowing beforehand all items in the sample.
The population mean (μ) is calculated using the expected value of the population variable (X).
The population variance (σ²) is calculated using the formula E[(x-μ)²]
The sample standard deviation (s) is calculated using the square root of s².

Development of a Sampling Distribution

An example illustrates how a sampling distribution is developed. This example uses a finite population (N = 4) and a random variable (X) that represents the age of individuals, where the ages are 18, 20, 22, and 24 years old.

Sampling Distribution of All Sample Means

Listing all possible samples of a given size (n=2) provides a sampling distribution.

Comparing the Population with its Sampling Distribution

The population data and its sampling distribution demonstrate that the sampling distribution mean is the same as the population mean, but the sampling distribution's standard deviation is smaller. (smaller standard error)

Sampling Distributions of Sample Means

The mean of a sampling distribution of sample means accurately reflects the population mean, regardless of the specific population's distribution.
Mean (x) is calculated, as the sum of individual means divided by the number of samples.

Variance of Sample Means

Sample variance (s²) is calculated considering both sampling with and without replacement.
Variance (x) decreases with larger sample sizes (n).

Rigorous Analysis for Random Sampling Without Replacement

Rigorous analysis involves formulas for population variance, incorporating the idea of sample covariance.

Finite Population Correction Factor

The finite population correction factor (N-n/N) is negligible when the population size is large compared to the sample size.

Sampling Distribution of Sample Means

If a population follows a normal distribution, the sample means will also follow a normal distribution.

Central Limit Theorem

The Central Limit Theorem (CLT) states that the sampling distribution of the sample means approaches a normal distribution as the sample size (n) increases, regardless of the population's distribution shape.

History of LLN and CLT

Details regarding the origins of the laws are provided, along with important contributors' names and affiliations.

Example Applying CLT

A practical calculation illustrating how the CLT can be used to calculate probabilities based on sample means in a large population.

CLT for Random Sampling Without Replacement

Discussion of CLT

Sampling Distribution of Sample Proportions

Sample proportions follow a binomial distribution.
As sample sizes increase, the sampling distribution of proportions approaches a normal distribution.

Example Applying Sampling Distributions of Sample Proportions

Calculating probabilities associated with sample proportions using the normal approximation.

Sampling Distributions of Sample Variances

Sample variances are essential for understanding how data vary around the sample mean.

Sampling Distribution of the Sample Variance

Sample variance (s²) is a natural estimate for population variance (σ²).
Sample variance's expected value (E[s²]) equals the population variance (σ²) for normally distributed populations.

x² Distribution

The x² distribution arises from independent standard normal random variables.
Its values are always positive.

Mean and Variance of the Sample Variance

Further Results

The expected value (E[s²]) and variance of s² are presented, along with conditions for unbiasedness.

Summary

The summary provides a table summarizing estimators for the mean (x ), proportion (p), and variance (s²) with considerations of sample size (n), population size (N), and whether sampling is with replacement or not.

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Description

This quiz explores the concepts of unbiased estimators and their efficiency in statistical analysis. It covers definitions, comparisons of variances, and specific conditions that determine efficiency among estimators. Dive into the details of relative efficiency, roles of sample mean versus sample median, and behavior as sample size increases.