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Sampling and Estimation Flashcards

1
Q

Simple Random Sampling

A

Selection of a sample such that each item of the population has the same likelihood of being included in the sample.

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2
Q

Systematic Sampling

A

Selection of every nth member from a population.

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3
Q

Sampling Error

A

The difference between a sample statistic (mean, variance, s-dev) and its corresponding population parameter.

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4
Q

Sampling Error of the mean

A
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5
Q

Sampling Distribution

A

Probability distribution of all possible sample statistics computed from a set of equal sized samples randomly selected from the same population.

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6
Q

Stratified random sampling

A

Use of a classification system to separate the population into smaller groups based on one or more distinguishing characteristics.

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7
Q

Time-series data

A

Observations taken over a period of time at specific and equally spaced time intervals.

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8
Q

Cross-sectional data

A

Sample of observations taken at a single point in time.

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9
Q

Longitudinal Data

A

Observatiosn over time of multiple characteristics of the same entity

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10
Q

Panel data

A

Observations over time of the same characteristic for multiple entities

The size of the samples from each strata is based on the relative size of the strata relative to the population and not necessarily same across population
Classify the population into smaller groups based on one or more distinguishing characteristics
Take a random sample from each subgroup and pooled together.
The size of sample from each subgroup is based on relative of the group

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11
Q

Central Limit Theorem

A

For simple random samples of size n from a population with a mean of m and a finite variance, the sampling distribution of the same mean approaches a normal probability distribution with mean m and a variance equal to variance/n

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12
Q

Point estimates

A

Single (sample) values used to estimate population parameters.

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13
Q

Confidence interval

A

Confidence intervals are usually constructed by adding or subtracting an appropriate value from the point estimate

* Point Estimate +_ Reliability factor x Standard Error*

Range of values within which the actual value of a parameter will lie, given the probability of 1 - a

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14
Q

Level of significance

A

α

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15
Q

Degree of confidence

A

1 - α

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16
Q

Confidence interval form

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17
Q

Desirable properties of an estimator

A

Unbiasedness, efficiency, and consistency.

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18
Q

Desirable properties of an estimator - definitions

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19
Q

Student’s t-distribution

A

Bell-shaped probability distribution that is symmetrical about its mean.

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20
Q

Properties of student’s t-distribution

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21
Q

Confidence interval for the population mean (normal distribution with a known variance)

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22
Q

Commonly used standard normal distribution reliability factors

A
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23
Q

Confidence intervals for a population mean that is normal with unknown variance

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24
Q

Criteria for selecting the appropriate test statistic

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25
Data Mining
Occurs when analysts use the same database to search fo rpatterns or trading rules until they discover one that "works."
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Data-mining bias
Results where the statistical significance of the pattern is overestimated because the results were found through data mining.
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Sample selection bias
When some data is systematically excluded from the analysis because of lack of availability.
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Survivorship bias
The most common bias... for example, when funds are no longer included because they have ceased to exist due to closure or merger.
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Look-ahead bias
When a relationship is tested using sample data that was not availabe on the test date. For example Consider a test of a trading rule that is based on the price to book value Stock Price Are available for all companies at the same point of time Book Value While the year end book values may not be available for all companies until 30 to 60 days after the fiscal year ends
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Time-period bias
Occurs when the time period over which the data was gathered is too short or too long. Too Short Results may reflect phenomenon specific to that time period or perhaps data mining Too Large The fundamental economic relationships that underlie the results may have changed
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Desirable Properties of Estimator
1: - Unbiasedness 2: - Efficient 3: - Consistent
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Unbiasedness
An unbiased estimator is one whose expected value equals the parameter it's intended to estimate. Expected Value An unbiased estimator is one for which the expected value of the estimator is equal to parameter you are trying to estimate For example because the expected value of the sample mean is equal to the population mean the sample mean is an unbiased estimator of the population mean
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Efficiency Enterprise Value
An estimator is efficient if no other estimator has a sample distribution with smaller variance. ## Footnote An estimator is efficient if the variance of its sampling distribution is smaller the all of other unbiased estimators of the parameter you are trying to estimate
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Consistency Sample Size Company Secretary
A consistent estimator is one for which the probability if estimates close to the value of the population parameter increases as sample size increases. A consistent estimator is as you increase the sample size the accuracy of the parameter estimator also increases. Therefore with an increase in the sample size the standard error of the sample mean also decrease and the sampling distribution bunches more closely around the population mean As the sample size approaches infinity the standard error approaches zero
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Confidence Interval
A 100(1-α)% confidence interval: Point estimate +/- Reliability factor x Standard error.
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Issues Regarding Selection of the Appropriate Sample Size
Limitations 1:-Larger samples may contain observations from a different from a different population (distribution) 2:-÷The cost of using a larger sample must be weighted against the value of the increase in the precision from the increase in sample size
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Larger Sample Size Advantages
Larger Sample Size Advantages 1: -Reduces sampling error and standard deviation of the sample statistics around its population value 2: -Confidence intervals are narrower when samples are larger and the standard errors of the point estimates of population parameter are less
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Sampling Error
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Size of Samples in Stratified Random Sampling
The size of the samples from each strata is based on the relative size of the strata relative to the population and not necessarily same across population Classify the population into smaller groups based on one or more distinguishing characteristics Take a random sample from each subgroup and pooled together. The size of sample from each subgroup is based on relative of the group
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What is the probability of confidence interval ?
1- alpha
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Alpha and 1 - Alpha
Confidence Interval estimates result in a range of values within which the actual of a parameter will lie given the probability of 1-alpha Here alpha is called the level of significance for confidence interval And the probability 1-alpha is referred to as the degree of confidence
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Normal Distribution Classification
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Non Normal Distribution Classification
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Normal Distribution Known Variance Small Sample Size
Z Statistics Interpretation **Probabilistic** After repeatedly taking samples of CFA candidates administering the practice exam and constructing confidence intervals for each sample's mean 99% of the resulting confidence intervals will in the long run include the population mean **Practical** We are 99% Confident that the population mean score is between 73.5 and 86.45 for candidates from this population
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Normal Distribution Known Variance Large Sample Size
Z Statistics Interpretation ***Probabilistic*** After repeatedly taking samples of CFA candidates administering the practice exam and constructing confidence intervals for each sample's mean 99% of the resulting confidence intervals will in the long run include the population mean ***Practical*** We are 99% Confident that the population mean score is between 73.5 and 86.45 for candidates from this population
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Normal Distribution UnKnown Variance Small Sample Size
***t-Statisitcs*** Owing to the relatively fatter tails of the t-distribution confidence intervals constructed using reliability t-reliability factors will be more conservative (wider) than those constructed using z-reliability factors Unlike the standard normal distribution the reliability factor for t-distribution depend on the sample size so we can't rely on commonly used set if reliability factors
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Normal Distribution Unknown Variance Large Sample Size
***t-statistics*** Owing to the relatively fatter tails of the t-distribution confidence intervals constructed using reliability t-reliability factors will be more conservative (wider) than those constructed using z-reliability factors Unlike the standard normal distribution the reliability factor for t-distribution depend on the sample size so we can't rely on commonly used set if reliability factors
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Non Normal Distribution Known Variance Small Sample Size
**NA**
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Non Normal Distribution Known Variance Large Sample Size
***Z Statistics*** If the distribution is non normal but the population variance is known the z statistics can be used as long as the sample size is large n is greater than 30 We do this because central limit theorem assures us that the distribution of the sample mean is approximately normal when the sample is large
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Non Normal Distribution UnKnown Variance Small Sample Size
***NA***
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Non Normal Distribution UnKnown Variance Large Sample Size
***t-Statistics*** It is also acceptable to use the z-statistics although use of the t-statistics is more conservative If the distribution is non normal but the population variance is known the t-statistics can be used as long as the sample size is large n is greater than 30
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Student's t-distribution
It is a bell shaped probability distribution that is symmetrical about its mean It is appropriate distribution to use when constructing confidence intervals based on small samples (n It may also be appropriate to use the t-distribution when population variance is unknown and sample size is large enough that the central limit theorem will assure that the sample distribution is approximately normal
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Student's t-distribution Properties
***Properties *** 1: - Symmetrical 2: -Defined by single parameter the degrees of freedom where the degrees of freedom are equal to the number of sample observation minus one for sample mean 3: -It has more probability in the tails (fatter tails) than normal distribution 4: -As the degrees of freedom (the sample size ) the shape of the t-distribution more closely approaches a standard normal distribution
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Standard Error
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Standard Deviation is Equal to
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Standard Error of the Sample Mean
The **_Positive Square Root_** _of the_ **_Variance_** of the Sample Statistics
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What does Standard Error Truely Signify
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Standard Deviation Formula for Square Root of Sum of Squared Deviation to the Mean
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Time Period Bias
Can result if the time period over which the data is gathered is either too short or too large Too Short:- Results may reflect phenomenon specific to that time period or perhaps data mining
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When to use sample standard deviation and when to use population standard deviation in calculating standard error
When population standard deviation is given use that to calculate standard error but when it's not given (population standard deviation) than use sample standard deviation.
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How to identify whether to use t-distribution or z - distribution
See the sample size See what is given standard deviation . Sample or population
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Formula Confidence Interval for Bernoulli Trial
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Formula for Width of Confidence Interval
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Properties of Central Limit Theorem
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