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Hypothesis Testing Steps in Hypothesis Testing: Two-Tailed Test (Z-test @ 5%) 1. State the hypotheses Null hypothesis: = 0 2. Identify the test statistic and its probability Alternative hypothesis: 0 distribution where 0 is the hypothesised mean 3. Specify the significance level 4. State the decision rule Rejection area Rejection area 5. Collect the data and perform the calculations 6. Make the statistical decision 1.96 SE 1.96 7. Make the economic or SE 0 investment decision One-Tailed Test (Z-test @ 5%) Rejection area Null hypothesis: 0 Alternative hypothesis: > 0 1.645 SE 0 Hypothesis Testing – Test Statistic & Errors Test Statistic: sample statistic - hypothesis ed value Test statistic standard error of the sample statistic Test Concerning a Single Mean Type I and Type II Errors X - μ0 Test statistic,Z or t • Type I error is rejecting the null sX when it is true. Probability = s significance level. sx n Use σ x if available • Type II error is failing to reject the null when it is false. • The power of a test is the Decision H0 true H0 false probability of correctly rejecting the null (i.e. rejecting the null when Do not reject null Correct Type II error it is false) Reject null Type I Correct error Hypothesis about Two Population Means Normally distributed populations and independent samples Examples of hypotheses: H0 : μ1 μ2 0 versus Ha : μ1 μ2 0 H0 : μ1 μ2 5 versus Ha : μ1 μ2 5 (x1 x2 ) (μ1 μ2 ) Test statistic,t H0 : μ1 μ2 0 versus Ha : μ1 μ2 0 standard error H0 : μ1 μ2 3 versus Ha : μ1 μ2 3 etc...... Population variances unknown but Population variances unknown and assumed to be equal cannot be assumed equal s2 s2 Standard error n1 n2 (n1 1)s1 2 (n2 1)s2 2 s12 s22 s Standard error n1 n2 2 n1 n2 s2 is a pooled estimator of the common variance Degrees of freedom = (n1 + n2 - 2) Hypothesis about Two Population Means Normally distributed populations and samples that are not independent - “Paired comparisons test” Possible hypotheses: H0 : μd μd0 versus Ha : μd μd0 d μd0 H0 : μd μd0 versus Ha : μd μd0 Test statistic sd H0 : μd μd0 versus Ha : μd μd0 Symbols and other formula Application 1 n • The data is arranged in paired d sample mean difference di n i1 observations μd0 hypothesized value of the difference • Paired observations are observations that are 2 sd sample variance of the sample differences di dependent because they have something in common s sd standard error of the mean difference d • E.g. dividend payout of n companies before and after a degrees of freedom n 1 change in tax law Hypothesis about a Single Population Variance Possible hypotheses: 2 2 H0 : σ 2 σ 0 versus Ha : σ 2 σ 0 Test statistic, χ 2 n 1s2 2 2 H0 : σ 2 σ 0 versus Ha : σ 2 σ 0 σ 02 2 2 H0 : σ 2 σ 0 versus Ha : σ 2 σ 0 Assuming normal population Symbols Chi-square distribution is asymmetrical and s2 = variance of the sample Obtained bounded below by 0 data from the Chi- 02 = hypothesized value of square Obtained from tables. (df, 1 the Chi-square the population variance tables. (df, /2) - /2 ) n = sample size Degrees of freedom = n – 1 NB: For one-tailed test use Lower Higher critical value critical value or (1 – ) depending on whether it is a right-tail or Reject H0 Fail to reject H0 Reject H0 left-tail test. Hypothesis about Variances of Two Populations Possible hypotheses: The convention is to always put the larger variance on top 2 2 2 2 H0 : σ1 σ2 versus Ha : σ1 σ2 2 2 2 2 2 H0 : σ1 σ2 versus Ha : σ1 σ2 s1 Test statistic,F 2 2 2 2 2 H0 : σ1 σ2 versus Ha : σ1 σ2 s2 Assuming normal populations Degrees of freedom: F Distributions Obtained from the are asymmetrical F-distribution table numerator = n1 - 1, and bounded below for: by 0 denominator = n2 - 1 - one tailed test /2 - two tailed test Critical value Fail to reject H0 Reject H0 Correlation Analysis Sample Covariance and Correlation Coefficient Scatter Plots n y x Xi X Yi Y x x x Sample covariance i1 x x n 1 x x x x Correlation coefficient measures the direction and x x x x extent of linear association between two variables x x x x x x covariance x, y x Sample correlatio n coefficien t, rx, y sx sy s sample standard deviation 1.0 rx , y 1.0 Testing the Significance of the Correlation Coefficient r n2 Test statistic t Set Ho: = 0, and Ha: ≠ 0 1 r 2 Reject null if |test statistic| > critical t Degrees of freedom = (n - 2) Parametric and nonparametric tests Parametric tests: Nonparametric tests: • rely on assumptions regarding the • either do not consider a particular distribution of the population, and population parameter, or • are specific to population • make few assumptions about the parameters. population that is sampled. All tests covered on the previous Used primarily in three situations: slides are examples of parametric • when the data do not meet tests. 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?) Linear Regression Basic idea: a linear relationship between two variables, X and Y. i Y b0 b1 X 1 i Note that the standard error of estimate (SEE) is in the same units as ‘Y’ and hence should be viewed relative to ‘Y’. Y, dependent variable Yi x Mean of i values = 0 i error term x or residual x x ˆ Yi x x x x x ˆ ˆ ˆ Yi b0 b1 X i x x x x x x Least squares regression finds the straight line that minimises x x x x εi2 ˆ ( sum of the squared errors,SSE) X, independent variable Xi The Components of Total Variation n 2 Total variation Y Y SST i i1 n 2 n n 2 i i Unexplained variation Y Y εi SSE ˆ Explained variation Yi Y SSR i1 i1 i1 ANOVA, Standard Error of Estimate & R2 Sum of squares regression (SSR) Sum of squared errors (SSE) Standard Error of Estimate n Sum of squares total (SST) ε ˆ i=1 i 2 SSE SEE = = n-2 n-2 Coefficient of determination R2 is the proportion of the total variation in y that is explained by Interpretation the variation in x When correlation is strong (weak, i.e. near to zero) 2SSR SST - SSE • R2 is high (low) R = = SST SST • Standard error of the estimate is low (high) Assumptions & Limitations of Regression Analysis Assumptions Limitations 1. The relationship between the 1. Regression relations change over dependent variable, Y, and the time (non-stationarity) independent variable, X, is linear 2. If assumptions are not valid, the 2. The independent variable, X, is not interpretation and tests of random hypothesis are not valid 3. The expected value of the error 3. When any of the assumptions term is 0 underlying linear regression are 4. The variance of the error term is violated, we cannot rely on the the same for all observations parameter estimates, test (homoskedasticity) statistics, or point and interval forecasts from the regression 5. The error term is uncorrelated across observations (i.e. no autocorrelation) 6. The error term is normally distributed

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posted: | 12/4/2011 |

language: | English |

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