Estimators and Sampling in Statistics

The video tutorial explains the concept of estimators in statistics, focusing on unbiased estimators. It introduces the idea of using a sample to estimate population parameters, such as the mean. The tutorial demonstrates that the sample mean is an unbiased estimator of the population mean. It also provides an example of a biased estimator using sample variance and explains the adjustment needed to make it unbiased by dividing by n-1 instead of n.

What is the primary purpose of an estimator in statistics?

• To determine the size of a population

• To estimate a population parameter based on a sample

• To provide a guess about a sample statistic

• To calculate the exact value of a population parameter

Why do statisticians use samples instead of entire populations?

• Because samples are more accurate

• Because populations are always too large to observe

• Because populations are not important in statistics

• Because samples are easier to manipulate

What symbol is commonly used to represent the population mean?

• μ

• σ

• x̄

• n

How is the sample mean (x̄) calculated?

• By adding the population mean to the sample size

• By dividing the sum of all sample values by the sample size

• By subtracting the smallest value from the largest value

• By multiplying the sum of all sample values by the sample size

What does it mean for an estimator to be unbiased?

• The expected value of the estimator equals the population parameter

• The estimator always equals the sample mean

• The estimator is always less than the population parameter

• The estimator is always greater than the population parameter

What is the expected value of an estimator?

• The minimum value of the estimator

• The maximum value of the estimator

• The average of all possible values of the estimator

• The sum of all possible values of the estimator

Why is the sample mean considered an unbiased estimator of the population mean?

• Because it is always greater than the population mean

• Because its expected value equals the population mean

• Because it is always equal to the population mean

• Because it is always less than the population mean

What is a biased estimator?

• An estimator that is always less than the population parameter

• An estimator whose expected value equals the population parameter

• An estimator whose expected value does not equal the population parameter

• An estimator that is always greater than the population parameter

How is the sample variance typically estimated?

• By multiplying the sum of squared deviations by n-1

• By multiplying the sum of squared deviations by n

• By dividing the sum of squared deviations by n

• By dividing the sum of squared deviations by n-1

Why is dividing by n-1 instead of n important for estimating variance?

• It decreases the variance

• It increases the variance

• It makes the estimator unbiased

• It makes the estimator biased