For a 5 significance level with numerator degrees of

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For a 5% significance level, with numerator degrees of freedom 3 = - ) 1 n ( x and denominator degrees of freedom 6 = - ) 1 n ( y the F-distribution critical value is: 4.76 = c F The calculated test statistic of 7.095 exceeds the critical value and therefore, there is evidence to reject the null hypothesis of equal variance in the two sample periods. The data suggests that the variance of sales is significantly higher in the period of active price competition. Econ 325 – Chapter 10.4 8 The graph below illustrates that the calculated test statistic of 7.095 is in the rejection region for a 5% level one-tailed test. PDF of ) 6 , 3 ( F 7.095 Fc=4.76 0 upper tail area = 0.05 d o not rejec t Reject 0 H
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Econ 325 – Chapter 10.4 9 Now approach a decision rule by calculating and interpreting a p-value for the test. The p-value of the test is the smallest significance level at which a null hypothesis can be rejected. In the above exercise, the null hypothesis was rejected at a 5% level and so the p-value must be smaller than 5%. An exact p-value is obtained as the upper tail area of the F-distribution probability density function to the right of the calculated test statistic: p-value ) F ( P ) 6 , 3 ( 7.095 > = With Microsoft Excel select Insert Function: numerator degrees of freedom FDIST(7.095, 3, 6) denominator degrees of freedom This returns the answer: p-value = 0.021 Econ 325 – Chapter 10.4 10 The calculated p-value falls between 0.01 and 0.05 . The interpretation is that, although the null hypothesis can be rejected at a 5% level, it is not rejected at a 1% significance level. This conclusion can be confirmed by checking the F-distribution tables. With 0.01 = α the critical value from the ) 6 , 3 ( F distribution is: 9.78 = c F The calculated test statistic of 7.095 is below the 1% level critical value to suggest that the null hypothesis of equal population variances in the two sample periods is not rejected. head2right Comment on methodology – in applied work the computer software used for the statistical analysis of the data can also be used for the calculation of p-values for test statistics. Therefore, hypothesis testing conclusions can rely on the interpretation of p-values. The statistical tables printed in textbook Appendices can have a useful role as a backup check of the test decision.
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Econ 325 – Chapter 10.4 11 xrhombus More Examples of Testing for Equal Population Variances in Two Independent Samples
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