Reading 12: Hypothesis Testing

Question 1

Hypothesis (Definition)

Answer 1

A hypothesis is a statement about the value of a population parameter developed for the purpose of testing a theory or belief

Question 2

Hypothesis Testing Procedures (Definition)

Answer 2

Used to determine whether a hypothesis is unreasonable and should be rejected

Question 3

Hypothesis Testing Procedures (Steps)

Answer 3

1. State the hypothesis
2. Select the appropriate test statistic
3. Specify the level of significance
4. State the decision rule regarding the hypothesis
5. Collect the sample and calculate the sample statistics
6. Make a decision regarding the hypothesis
7. Make a decision based on the results of the test

Question 4

The Null Hypothesis

Answer 4

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

Question 5

The Alternative Hypothesis

Answer 5

What is concluded if there is enough evidence to reject the null hypothesis

Question 6

One-tailed test

Answer 6

If the return (x) is greater than zero (or vice versa)

Question 7

Two-tailed test

Answer 7

If the return (x) is simply different from zero

Uses two critical values (Rejection points)

Question 8

General decision rule for a two-tailed test

Answer 8

Reject Ho if: test statistic > upper critical value; or

test statistic < lower critical value

Question 9

Test Statistic

Answer 9

Difference between the sample statistic (point estimate) and the hypothesized value (Ho) scaled by the standard error of the sample statistic

Question 10

Test Statistic (Potential Distributions)

Answer 10

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

Question 11

Type I Error

Answer 11

Rejection of the null hypothesis when it is actually true

Question 12

Type II Error

Answer 12

The failure to reject the null hypothesis when it is actually false

Question 13

Decision Rule

Answer 13

If the test statistic is (greater, less than) the critical value, reject the null

Question 14

The Power of a Test (Definition)

Answer 14

Probability of correctly rejecting the null hypothesis when it is false

Useful for comparing multiple test statistics

Question 15

The Power of a Test (Formula)

Answer 15

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

Question 16

Significance Level

Answer 16

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

Question 17

Relationship between confidence interval and significance level

Answer 17

Confidence interval and level of significance are linked as level of significance = 1 – confidence interval

Question 18

Factors to consider when comparing statistical significant result with economic meaningful result

Answer 18

Statistical significance does not necessarily imply economic significance. Need to consider:

1. Transaction Costs
2. Taxes
3. Risks

Question 19

P – value

Answer 19

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

Question 20

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

Answer 20

t - distribution

Formula:

Test statistic = (sample mean - null hypothesis)
/ standard error (using sample standard distribution)

Question 21

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

Answer 21

z - distribution

Formula:

Test statistic = (sample mean - null hypothesis)
/ standard error (using population standard distribution)

Question 22

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

Answer 22

t - test

Population variances are assumed to be equal and sample observations are pooled to get the standard deviation

Question 23

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

Answer 23

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

Question 24

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

Answer 24

Paired Comparison Test

Tests whether the means of the difference between observations for the two samples are different. Dependence in influenced by:

1. Market returns
2. Economic conditions

Formula is: (sample mean difference + hypothesized mean of the difference which is usually zero) / standard error of the difference

Question 25

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)

Answer 25

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

Question 26

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

Answer 26

F - Test

Variance of the sample of n1, observations drawn from Population 1 / variance of the sample of n2, observations drawn from Population 2

Question 27

F-distributed test statistic (Characteristics)

Answer 27

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

Question 28

Parametric Test (Definition)

Answer 28

Rely on assumptions regarding the distribution of the population and are specific to population parameters (e.g. mean etc.)

Question 29

Nonparametric Test (Definition)

Answer 29

Either:

1. Do not consider a particular population parameter; or
2. Have few assumptions about the population that is sampled
3. Used for ranked observations (not suitable for parametric tests)

Question 30

Nonparametric Test (Occasions)

Answer 30

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

Question 31

Spearman Rank Correlation test (Definition and application)

Answer 31

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