Hypothesis Testing

A hypothesis is a statement made about a population parameter. It involves the process to test whether the statement is true or not at certain confidence level.

7-Steps of hypothesis testing:

1. State the hypothesis
2. Identify test statistic and its probability distribution
3. Specify significance level
4. State decision rule
5. Collect sample & Calculate test statistic
6. Make statistical decision
7. Make economic decision

Sample size and Potential mistake in sampling

Sample size
Increasing sample size benefits increase the confidence and reliability of the confidence interval, and thus the precision with which the population parameter can be estimated.

Factors that make larger sample size undesirable

• Additional expenses
• Population parameters have a tendency to change over time

Potential mistakes in sampling:
Data-snooping bias

• Conclusions of one analyst are guided by conclusions of others.

Data-mining bias

• Analyst keeps searching for patterns & trading rules until he find one that matches the data.

Sample selection bias

• Refer to the tendency to exclude a certain part of a population simply because the data is not
  available.

Survivorship bias

• Only funds/stocks that have survived to date are included. Big problem with stock indices.

Look-ahead bias

• Based on information that do not actually exist at the time of analysis, just based on analyst’s
  assumptions.

Time-period bias

Conclusions may apply only to a specific time period and are not repeatable over longer
time periods. Either too short or too long.

t-statistic vs. z-statistic

t-statistic:
It can be used when the population variance is unknown for large samples and small samples with normally distributed population. Comparing the t-statistic with critical value that comes from Student’s t table.

Critical value from Student’s t table
Look up df=n-1, p=significance for 1-sided test or p=significance/2 for 2-sided test

t-distribution is less peaked and flater tails than normal distribution but move closer at higher degrees of freedom.

z-statistic:
It is based on the standard normal distribution and can be used when population is normally distributed with known variance; also for large samples with unknown variance.

Summary of the applications of z-statisitc and t-distribution

My tips:

• Memorize the following commonly used values (from z-table):
• Confidence Interval Reliability factor
  90% 1.65 ; 95% 1.96 ; 99% 2.58
• Be able to look up Student's table given degrees of freedom and probability in the right tail.
  
  
  
  
  

Reliability Factor, Degree of freedom (df)

Reliability factor
For small samples from a population with unknown variance, reliability factor should be looked up from Student's t table {Look up df=n-1, p=significance/2 = (1-confidence)/2}.

For known variance or for large samples, you can use z-statistic.

Degree of freedom (df)
Used to determine the reliability factor of the confidence interval with the t–distribution. Degrees of freedom = n – 1, where n is the sample size.

Points estimators vs. Confidence interval (CI)

Point estimators
Use single(sample) values to estimate population paramenters

Confidence interval
Estimate range of value within which the actual parameters will lie, given the probability of 1-α. (1- α) is the degree of confidence.

Desirable properties of an estimate

• Lack of bias, i.e. expected value of estimator = true value of population parameter)
• Efficiency, i.e. low variance
• Consistency, i.e. accuracy must increase with sample size
  

Construct Confidence Intervals
The general structure for a (1 – α) confidence interval is given by:
CI= Point estimate ± Reliability factor x Standard error.

Standard error of the sample mean

This refers to the standard deviation of the sample statistic.

If the population standard deviation (σ) is given, standard error is calculated as:

Standard error =σ/(n)^1/2

If population standard deviation is unknown, the sample standard deviation (s) is used:

Standard error = s/(n)^1/2

Central limit theorem

If sample size is large (e.g. n≧30), 3 important characteristics for sampling distribution by using random draws, regardless of the population’s distribution,

• The sample mean will be approximately normally distributed.
• Mean of sample means = Mean of the population.
• Variance of sample means = Variance of population / Sample size, i.e. σ²/n
  

Notes:

• For large sample size, if population variance is unkown, use sample sampling error of the sample mean as the variance.
• Increase in sample size leads to decrease in variance of sample means

Time-series data, Cross-sectional data, Panel data

bleTime-series data
Refer to one variable taken over discrete, equally spaced periods of time.

Cross-sectional data
Focus on one period of time and measures a particular variable across several companies or industries, i.e.several variables at a single point in time.

Panel data
Involves several variables over time.

Simple random sampling, Stratified sampling, Sampling error, Sampling distribution

A data sample, or subset of a larger population, is used to help understand the behavior and characteristics of the entire population.

Simple random sampling
Pick random elements from a population, assuming every element of the population must have an equal chance of being selected.

Stratified sampling
Divide the population (strata) according to their attributes and characteristics, then pick random elements from each division. The subsets of the strata are then pooled to form a random sample.

Sampling error
The difference between sample statistics and population parameters

Sampling distribution
Probability distribution of all possible sample statistics