Reading 12: Hypothesis Testing Flashcards
Hypothesis (Definition)
A hypothesis is a statement about the value of a population parameter developed for the purpose of testing a theory or belief
Hypothesis Testing Procedures (Definition)
Used to determine whether a hypothesis is unreasonable and should be rejected
Hypothesis Testing Procedures (Steps)
- State the hypothesis
- Select the appropriate test statistic
- Specify the level of significance
- State the decision rule regarding the hypothesis
- Collect the sample and calculate the sample statistics
- Make a decision regarding the hypothesis
- Make a decision based on the results of the test
The Null Hypothesis
The null hypothesis is the hypothesis that the researcher wants to reject (Ho). It is generally a simple statement about the population parameter
Usually includes an ‘equal to’ condition
The Alternative Hypothesis
What is concluded if there is enough evidence to reject the null hypothesis
One-tailed test
If the return (x) is greater than zero (or vice versa)
Two-tailed test
If the return (x) is simply different from zero
Uses two critical values (Rejection points)
General decision rule for a two-tailed test
Reject Ho if: test statistic > upper critical value; or
test statistic < lower critical value
Test Statistic
Difference between the sample statistic (point estimate) and the hypothesized value (Ho) scaled by the standard error of the sample statistic
Test Statistic (Potential Distributions)
Because it is a random variable it has a set of potential distributions. These include:
t – distribution
z – distribution (standard normal distribution)
chi – square distribution
f – distribution
Type I Error
Rejection of the null hypothesis when it is actually true
Type II Error
The failure to reject the null hypothesis when it is actually false
Decision Rule
If the test statistic is (greater, less than) the critical value, reject the null
The Power of a Test (Definition)
Probability of correctly rejecting the null hypothesis when it is false
Useful for comparing multiple test statistics
The Power of a Test (Formula)
1 – P(Type II Error)
Can decrease the size of a type II error and increase the power of the test, only by increasing the sample size
Significance Level
Probability of making a Type I error and is designated by alpha e.g. a significance level of 5% means that there is a 5% chance that you will reject the ‘true’ null
You need the significance levels to determine the critical values needed to evaluate the test statistic
Relationship between confidence interval and significance level
Confidence interval and level of significance are linked as level of significance = 1 – confidence interval
Factors to consider when comparing statistical significant result with economic meaningful result
Statistical significance does not necessarily imply economic significance. Need to consider:
- Transaction Costs
- Taxes
- Risks
P – value
Probability of obtaining a test statistic ((sample state – hypothesis value)/standard error) that leads to a rejection of a null hypothesis, assuming that the null hypothesis is true
Smallest level of significance for which a null hypothesis can be rejected
What test statistic do you use when looking at a hypothesis test concerning:
1) Population mean of large and small samples
2) Normal Distribution
3) Unknown Variance
Please also include formula to determine the test statistic for this distribution
t - distribution
Formula:
Test statistic = (sample mean - null hypothesis)
/ standard error (using sample standard distribution)
What test statistic do you use when looking at a hypothesis test concerning:
1) Population mean of large and small samples
2) Normal Distribution
3) Known Variance
Please also include formula to determine the test statistic for this distribution
z - distribution
Formula:
Test statistic = (sample mean - null hypothesis)
/ standard error (using population standard distribution)
What test statistic do you use when looking at a hypothesis test concerning:
1) Equality of population means of two approximately normally distributed populations
2) Based on independent random samples
3) Equal variances
Please also include description of treatment of variances
t - test
Population variances are assumed to be equal and sample observations are pooled to get the standard deviation
What test statistic do you use when looking at a hypothesis test concerning:
1) Equality of population means of two approximately normally distributed populations
2) Based on independent random samples
3) Unequal variances
Please also include description of treatment of variances
t - test
No assumption is made regarding variances and it uses an approximated value for the degrees of freedom
Both sample variances are used to get the standard error of the difference in means when we assume the population variances are not equal
What test statistic do you use for a hypothesis test concerning:
1) mean difference of two normally distributed populations
2) based on dependent random samples
Please explain what causes dependence as well as the formula of the test
Paired Comparison Test
Tests whether the means of the difference between observations for the two samples are different. Dependence in influenced by:
- Market returns
- Economic conditions
Formula is: (sample mean difference + hypothesized mean of the difference which is usually zero) / standard error of the difference
What test statistic do you use for a hypothesis test concerning:
1) variance of a normally distributed population
Please include an explanation of the test as well as the formula for the test (note the symmetry of the distribution)
Chi Square Test
Used for hypothesis tests concerning the variance of a normally distributed population. Note that there is no mention of means in this test.
The distribution is asymmetrical with the entire distribution falling to the right of ‘0’
Formula is: ((n-1)sample variance (squared)) / hypothesized value for the population variance
What test statistic do you use for a hypothesis test concerning:
1) The equality of variances of two normally distributed populations
2) based on two independent random variables
Please include the name and formula for the test
F - Test
Variance of the sample of n1, observations drawn from Population 1 / variance of the sample of n2, observations drawn from Population 2
F-distributed test statistic (Characteristics)
Right skewed and bounded by zero of the left
Shape determined by two separate degrees of freedom
When variances are equal, the value of the test statistic is 1
Upper critical value is always greater than 1 so we only consider that side
Parametric Test (Definition)
Rely on assumptions regarding the distribution of the population and are specific to population parameters (e.g. mean etc.)
Nonparametric Test (Definition)
Either:
- Do not consider a particular population parameter; or
- Have few assumptions about the population that is sampled
- Used for ranked observations (not suitable for parametric tests)
Nonparametric Test (Occasions)
Assumptions about the distribution of the random variable that support a parametric test are not met
Hypothesis test of the mean that comes from small sample size or non-normal distribution
In this case the t – distribution and z – distribution would not be appropriate
When data ranks e.g. an ordinal measurement scale rather than values
Spearman Rank Correlation test (Definition and application)
Used when non-normal distribution
Measures the degree of correlation between two variables e.g. that high returns in Y1 will be high in Y2.
Negative spearman rank indicates that high returns in Y1 will instead be low in Y2
