chap10S4PRN

# Probabilities of 005 or 001 therefore when testing

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probabilities of 0.05 or 0.01 . Therefore, when testing against a two- sided alternative, the information in the table offers a choice of the significance level as either: 0.10 = α (a 10 % level) or 0.02 = α (a 2 % level). In applied work, computer software is used for the calculation of descriptive statistics, the test statistic and an accompanying p-value. A p-value for the two-tailed test is calculated as: p-value > = - - 2 y 2 x ) 1 n , 1 n ( s s F P 2 y x For a chosen significance level α , the decision rule is to reject the null hypothesis if: p-value < α

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Econ 325 – Chapter 10.4 5 A one-sided alternative can be used as follows. Test the null hypothesis: 2 Y 2 X 0 : H σ = σ or 2 Y 2 X 0 : H σ σ against the alternative: 2 Y 2 X 1 : H σ > σ If a comparison of the calculated sample variances shows 2 y 2 x s s < then there is no evidence to reject the null hypothesis. On the other hand, if 2 y 2 x s s > then the null hypothesis may or may not be rejected. To make a decision, calculate a test statistic as the variance ratio: 2 y 2 x s s For a chosen significance level α , the decision rule is to reject the null hypothesis if: c 2 y 2 x F s s > where c F is the F-distribution critical value that satisfies: α = > - - ) F F ( P c ) 1 n , 1 n ( y x Econ 325 – Chapter 10.4 6 Example Annual total sales of a company are reported for two sample periods: Period 1: active price competition in the industry (4 years) Period 2: price collusion in the industry (7 years) It is hypothesized that total sales should vary more in an industry with active price competition compared to a market with price collusion. Denote 2 X σ and 2 Y σ as the population variances in the two sample periods. Test the null hypothesis: 2 Y 2 X 0 : H σ = σ against the alternative: 2 Y 2 X 1 : H σ > σ From the data set, the sample statistics are: Period 1: 4 = x n 114.09 = 2 x s 10.68 = x s Period 2: 7 = y n 16.08 = 2 y s 4.01 = y s
Econ 325 – Chapter 10.4 7 The test statistic is the variance ratio: 7.095 16.08 114.09 = = 2 y 2 x s s This can be compared with the critical values reported in the F-distribution Appendix Table.

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