That is we reject the hypothesis that there is no l ti hi b t i d f d dit

# That is we reject the hypothesis that there is no l

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That is, we reject the hypothesis that there is no l ti hi b t i d f d dit relationship between income and food expenditure, and conclude that there is a statistically significant positive relationship between household income Principles of Econometrics, 4t h Edition Page 30 Chapter 3: Interval Estimation and Hypothesis Testing positive relationship between household income and food expenditure 3.4 Examples of Hypothesis Tests Example: right-tail test 3.4.1b One-tail Test of an Economic Hypothesis The null hypothesis is H 0 : β 2 5.5 The alternative hypothesis is H 1 : β 2 > 5 5 The alternative hypothesis is 2 > 5.5 The test statistic is t = ( b 2 - 5.5)/se( b 2 ) ~ t ( N – 2) if the null hypothesis is true Select α = 0.01 The critical value for the right-tail rejection region is the 99 th percentile of the t- distribution with N 2 = 38 degrees of freedom, t (0.99,38) = 2.429 Th ill j t th ll h th i if th Thus we will reject the null hypothesis if the calculated value of t 2.429 If t < 2 429 we will not reject the null hypothesis Principles of Econometrics, 4t h Edition Page 31 Chapter 3: Interval Estimation and Hypothesis Testing If < 2.429, we will not reject the null hypothesis 3.4 Examples of Hypothesis Tests Using the food expenditure data, we found that b 2 = 10.21 with standard error se( b 2 ) = 2.09 3.4.1b One-tail Test of an Economic Hypothesis 10.21 with standard error se( ) 2.09 The value of the test statistic is: 2 5.5 10.21 5.5 b Since t = 2 25 < 2 429 we do not reject the null 2 2.25 se 2.09 t b Since = 2.25 < 2.429 we do not reject the null hypothesis that β 2 5.5 The profitability of a new supermarket required The profitability of a new supermarket required β 2 > 5.5. Hence we are not able to conclude that the new supermarket will be profitable and will Principles of Econometrics, 4t h Edition Page 32 Chapter 3: Interval Estimation and Hypothesis Testing not begin construction 3.4 Examples of Hypothesis Tests Example: left-tail test 3.4.2 Left-tail Tests The null hypothesis is H 0 : β 2 15 The alternative hypothesis is H 1 : β 2 < 15 The alternative hypothesis is 2 The test statistic is t = ( b 2 - 15)/se( b 2 ) ~ t ( N – 2) if the null hypothesis is true Select α = 0.05 The critical value for the left-tail rejection region is the 5 th percentile of the t- distribution with N – 2 = 38 degrees of freedom, t (0.05,38) = -1.686. Th ill j t th ll h th i if th Thus we will reject the null hypothesis if the calculated value of t -1.686 If t > -1 686 we will not reject the null hypothesis Principles of Econometrics, 4t h Edition Page 33 Chapter 3: Interval Estimation and Hypothesis Testing If > -1.686, we will not reject the null hypothesis 3.4 Examples of Hypothesis Tests Using the food expenditure data, we found that b 2 = 10.21 with standard error se( b 2 ) = 2.09 3.4.2 Left-tail Tests 10.21 with standard error se( ) 2.09 The value of the test statistic is: 2 5.5 10.21 15 b Since t = 2 29 < 1 686 we reject the null 2 2.29 se 2.09 t b   Since = -2.29 < -1.686 we reject the null hypothesis that β 2 15 and accept the alternative that β 2 < 15 We conclude that households spend less than \$15 from each additional \$100 income on food Principles of Econometrics, 4t h Edition Page 34 Chapter 3: Interval Estimation and Hypothesis Testing   • • • 