Sampling And Estimation

Question 0

Sampling plan

Answer 0

The set of rules used to select a sample.

Question 1

Parameter

Answer 1

A descriptive measure computed from or used to describe a population of data, conventionally represented by Greek letters.

Question 2

Simple random sample

Answer 2

A subset of a larger population created in such a way that each element of the population has an equal probability of being selected to the subset.

Question 3

Systematic sampling

Answer 3

A procedure of selecting every kth member until reaching a sample of the desired size. The sample that results from this procedure should be approximately random.

Question 4

Sampling error

Answer 4

The difference between the observed value of a statistic and the quantity it is intended to estimate.

Question 5

Stratified random sampling

Answer 5

In stratified random sampling, the population is divided into subpopulations (strata) based on one or more classification criteria. Simple random samples are then drawn from each stratum in sizes proportional to the relative size of each stratum in the population. These samples are then pooled to form a stratified random sample.

Question 6

Indexing

Answer 6

An investment strategy in which an investor constructs a portfolio to mirror the performance of a specified index.

Question 7

Monetary policy

Answer 7

Actions taken by a nation’s central bank to affect aggregate output and prices through changes in bank reserves, reserve requirements, or its target interest rate.

Question 8

Sharpe ratio

Answer 8

The average return in excess of the risk-free rate divided by the standard deviation of return; a measure of the average excess return earned per unit of standard deviation of return.

Question 9

Central limit theorem

Answer 9

Given a population described by any probability distribution having mean μ and finite variance σ2, the sampling distribution of the sample mean X ( x bar*) computed from samples of size n from this population will be approximately normal with mean μ (the population mean) and variance σ2/n (the population variance divided by n) when the sample size n is large.

Question 10

Standard error of the sample mean

Answer 10

For sample mean X⎯⎯⎯ calculated from a sample generated by a population with standard deviation σ, the standard error of the sample mean is given by one of two expressions:

Equation (1) 

σX⎯⎯⎯=σ / √n

when we know σ, the population standard deviation, or by

Equation (2) 

sX⎯⎯⎯= s /√n

when we do not know the population standard deviation and need to use the sample standard deviation, s, to estimate it.6

In practice, we almost always need to use Equation 2. The estimate of s is given by the square root of the sample variance, s2, calculated as follows:

Equation (3) 
2 2
s =∑(Xi−X⎯⎯⎯) / n−1

Question 11

Properties of the distribution of the sample mean

Answer 11

The distribution of the sample mean X⎯⎯⎯ will be approximately normal.

The mean of the distribution of X⎯⎯⎯ will be equal to the mean of the population from which the samples are drawn.

The variance of the distribution of X⎯⎯⎯ will be equal to the variance of the population divided by the sample size.

Question 12

Estimator

Answer 12

An estimation formula; the formula used to compute the sample mean and other sample statistics are examples of estimators.

Question 13

Point estimate

Answer 13

A single numerical estimate of an unknown quantity, such as a population parameter.

Question 14

Unbiased estimator

Answer 14

An unbiased estimator is one whose expected value (the mean of its sampling distribution) equals the parameter it is intended to estimate.

Question 15

Efficiency of an unbiased estimator

Answer 15

An unbiased estimator is efficient if no other unbiased estimator of the same parameter has a sampling distribution with smaller variance.

Question 16

Consistency of an estimator

Answer 16

A consistent estimator is one for which the probability of estimates close to the value of the population parameter increases as sample size increases.

Question 17

Confidence interval

Answer 17

Definition of Confidence Interval. A confidence interval is a range for which one can assert with a given probability 1 − α, called the degree of confidence, that it will contain the parameter it is intended to estimate. This interval is often referred to as the 100(1 − α)% confidence interval for the parameter.

Question 18

Construction of confidence intervals

Answer 18

A 100(1 − α)% confidence interval for a parameter has the following structure.

Point estimate ± Reliability factor × Standard error
where

Point estimate = a point estimate of the parameter (a value of a sample statistic)

Reliability factor = a number based on the assumed distribution of the point estimate and the degree of confidence (1 − α) for the confidence interval

Standard error = the standard error of the sample statistic providing the point estimate13

Question 19

Confidence Intervals for the Population Mean (Normally Distributed Population with Known Variance)

Answer 19

A 100(1 − α)% confidence interval for population mean μ when we are sampling from a normal distribution with known variance σ2 is given by

X⎯ ± z(α/2) σ/√n

Question 20

Reliability Factors for Confidence Intervals Based on the Standard Normal Distribution

Answer 20

We use the following reliability factors when we construct confidence intervals based on the standard normal distribution:

90 percent confidence intervals: Use z0.05 = 1.65

95 percent confidence intervals: Use z0.025 = 1.96

99 percent confidence intervals: Use z0.005 = 2.58

Question 21

Confidence Intervals for the Population Mean—The z-Alternative (Large Sample, Population Variance Unknown)

Answer 21

A 100(1 − α)% confidence interval for population mean μ when sampling from any distribution with unknown variance and when sample size is large is given by

X⎯ ± zα/2 s/√n

Question 22

Degrees of freedom (df)

Answer 22

The number of independent observations used.

Question 23

Confidence Intervals for the Population Mean (Population Variance Unknown)—t-Distribution.

Answer 23

If we are sampling from a population with unknown variance and either of the conditions below holds:

the sample is large, or

the sample is small but the population is normally distributed, or approximately normally distributed,

then a 100(1 − α)% confidence interval for the population mean μ is given by 

X⎯±tα/2 s/√n

where the number of degrees of freedom for tα/2 is n − 1 and n is the sample size.

Question 24

Data mining

Answer 24

The practice of determining a model by extensive searching through a dataset for statistically significant patterns. Also called data snooping.

Question 25

Out of sample test

Answer 25

The practice of determining a model by extensive searching through a dataset for statistically significant patterns. Also called data snooping.

Question 26

Intergenerational data mining

Answer 26

A form of data mining that applies information developed by previous researchers using a dataset to guide current research using the same or a related dataset.

Question 27

Sample selection bias

Answer 27

Bias introduced by systematically excluding some members of the population according to a particular attribute—for example, the bias introduced when data availability leads to certain observations being excluded from the analysis.

Question 28

Survivorship bias

Answer 28

The bias resulting from a test design that fails to account for companies that have gone bankrupt, merged, or are otherwise no longer reported in a database.

Question 29

Look ahead bias

Answer 29

A bias caused by using information that was unavailable on the test date.

Question 30

Time period bias

Answer 30

The possibility that when we use a time-series sample, our statistical conclusion may be sensitive to the starting and ending dates of the sample.