1. Hypothesis Testing Steps in Hypothesis Testing:
1. State the hypotheses
2. Identify the test statistic and its probability distribution
3. Specify the significance level
4. State the decision rule
5. Collect the data and perform the calculations
6. Make the statistical decision
7. Make the economic or investment decision
Two-Tailed Test 5%)
Null hypothesis:  = 0
Alternative hypothesis:   0
where 0 is the hypothesised mean
1.96 SE 0
Rejection area
Rejection area
One-Tailed Test 5%)
Rejection area
Null hypothesis:   0
Alternative hypothesis:  > 0
1.645 SE 0

2. Hypothesis Testing – Test Statistic & Errors
Test Concerning a Single Mean
Type I and Type II Errors
Type I error is rejecting the null when it is true. Probability = significance level.
Type II error is failing to reject the null when it is false.
The power of a test is the probability of correctly rejecting the null (i.e. rejecting the null when it is false)
Do not reject null
Reject null
Correct
Type I error
Type II error
Decision
H0 true
H0 false

3. Hypothesis about Two Population Means
Normally distributed populations and independent samples
Examples of hypotheses:
Population variances unknown but assumed to be equal
s2 is a pooled estimator of the common variance
Degrees of freedom = (n1 + n2 - 2)
Population variances unknown and cannot be assumed equal

4. Hypothesis about Two Population Means
Normally distributed populations and samples that are not independent -
“Paired comparisons test”
Possible hypotheses:
Symbols and other formula
Application
The data is arranged in paired observations
Paired observations are observations that are dependent because they have something in common
E.g. dividend payout of companies before and after a change in tax law

5. Hypothesis about a Single Population Variance
Possible hypotheses:
Assuming normal population
Symbols
s2 = variance of the sample data
02 = hypothesized value of the population variance
n = sample size
Degrees of freedom = n – 1
NB: For one-tailed test use  or (1 – ) depending on whether it is a right-tail or left-tail test.
Chi-square distribution is asymmetrical and bounded below by 0
Obtained from the Chi-square tables. (df, 1 - /2 )
Obtained from the Chi-square tables. (df, /2)
Lower
critical value
Higher
critical value
Reject H0
Fail to reject H0
Reject H0

6. Hypothesis about Variances of Two Populations
Possible hypotheses:
Assuming normal populations
The convention is to always put the larger variance on top
Degrees of freedom:
numerator = n1 - 1,
denominator = n2 - 1
F Distributions are asymmetrical and bounded below by 0
Obtained from the F-distribution table for:
 - one tailed test
/2 - two tailed test
Critical
value
Fail to reject H0
Reject H0

7. Correlation Analysis Sample Covariance and Correlation Coefficient
Correlation coefficient measures the direction and extent of linear association between two variables
Scatter Plots x y x
Testing the Significance of the Correlation Coefficient
Set Ho:  = 0, and Ha:  ≠ 0
Reject null if |test statistic| > critical t
Degrees of freedom = (n - 2)

8. Parametric and nonparametric tests
rely on assumptions regarding the distribution of the population, and
are specific to population parameters.
All tests covered on the previous slides are examples of parametric tests.
Nonparametric tests:
either do not consider a particular population parameter, or
make few assumptions about the population that is sampled.
Used primarily in three situations:
when the data do not meet distributional assumptions
when the data are given in ranks
when the hypothesis being addressed does not concern a parameter (e.g. is a sample random or not?)

9. Linear Regression Yi Xi
Basic idea: a linear relationship between two variables, X and Y.
Note that the standard error of estimate (SEE) is in the same units as ‘Y’ and hence should be viewed relative to ‘Y’. x
Y, dependent variable Xi
i error term or residual
X, independent variable Yi
Mean of i values = 0
Least squares regression finds the straight line that minimises

10. The Components of Total Variation

11. ANOVA, Standard Error of Estimate & R2
Sum of squares regression (SSR)
Sum of squared errors (SSE)
Sum of squares total (SST)
Standard Error of Estimate
Coefficient of determination
R2 is the proportion of the total variation in y that is explained by the variation in x
Interpretation
When correlation is strong (weak, i.e. near to zero)
R2 is high (low)
Standard error of the estimate is low (high)

12. Assumptions & Limitations of Regression Analysis
The relationship between the dependent variable, Y, and the independent variable, X, is linear
The independent variable, X, is not random
The expected value of the error term is 0
The variance of the error term is the same for all observations (homoskedasticity)
The error term is uncorrelated across observations (i.e. no autocorrelation)
The error term is normally distributed
Limitations
Regression relations change over time (non-stationarity)
If assumptions are not valid, the interpretation and tests of hypothesis are not valid
When any of the assumptions underlying linear regression are violated, we cannot rely on the parameter estimates, test statistics, or point and interval forecasts from the regression