【正文】 2　 find out its sampling distribution3　 choose a level of significance α4　 determine the critical value (s) of the test statistic at the chosen level of significance5　 pare the value of the test statistic obtained from the sample at hand with the critical value (s)6　 reject the null hypothesis if the puted value of the test statistic exceeds the critical value (s)1 if the test statistic has a negative value, we consider its absolute value and say that if the absolute value of the test statistic exceeds the critical value, we reject the null hypothesis.1 We can find the p value of the test statistic and reject the null hypothesis if the p value is smaller than the chosen αvalue求得統(tǒng)計量的p值，如果p值小于顯著水平α，則拒絕零假設 Testing the joint hypothesis that B2=B3=0 or R2=0：檢驗聯(lián)合假設Null hypothesis1　 This null hypothesis is a joint hypothesis that B2 and B3 are jointly or simultaneously equal to ，即B2,B3聯(lián)合或同時為令（而不是單獨為零）2　 This hypothesis states that the two explanatory variables together have no influence on 。3　 This is the same as saying that等同于2 Temptationl The temptation here is to state that since individually b2 and b3 are statistically different fromzero in the present example, then jointly or collectively they also must be statistically different from zero, that we reject the null ，既然b2，b3各自均顯著不為零，那么它們一定也聯(lián)合或集體顯著不為零，即拒絕這個零假設l In other words, since age of the antique clock and the number of bidders at the auction, eachhas a significant effect on the auction price, together they also must have a significant effect on the auction ，那么它們一起也一定會對拍賣價格有顯著影響l When multicollineratiy exists, in a multiple regression one ore more variables individually haveno effect on the dependent variable but collectively they have a significant impact on ，一個或多個解釋變量各自對應變量沒有影響，但卻聯(lián)合對應變量有影響。l This means that the ttesting procedure discussed previously, although valid for testing thestatistical significance of an individual regression coefficient, is not valid for testing the joint ，但對于聯(lián)合假設卻是無效的。2 F test statistic（會考小題10分）l F follows F distribution with 2 and (n3) . in the numerator and denominator, respectively.服從分子自由度為2，分母自由度為（n3）的F分布l In general, if the regression model has k explanatory variables including the intercept term, theF ratio has (k1) ., in the numerator and (nk) . in the ，如果回歸模型有k個解釋變量（包括截距），則F值的分子自由度為（k1），分母自由度為（nk）l How can we use the F ratio to test the joint hypothesis that both X2 and X3 have no impact onY ?如何利用給出的F值檢驗聯(lián)合假設：X2和X3對Y沒有影響呢？The answer is evident. If the numerator is larger the its denominator. If the variance of Y explained by the regression (. by X2 and X3) is larger than the variance not explained by the regression. The F ratio is greater than ，即如果Y由回歸解釋的部分（即由X2和X3解釋部分）比未被回歸結實的部分大，則F值將大于1。l Therefore, as the variance explained by the X variables bees increasingly larger, relative tothe unexplained variance, the F ratio will be increasingly larger, too. 因此，隨著解釋變量對應變量Y變異的解釋比例逐漸增大，F(xiàn)值也將逐漸增大。l Thus an increasingly large F ratio will be evidence against the null hypothesis that the two (ormore) explanatory variables have no effect on Y. 因此，F(xiàn)值越大，則拒絕零假設的理由越充分：兩個（或多個）解釋變量對應變量Y無影響。l We pare this puted F value with the critical F value for 2 and (n3) . at the chosenlevel of α, the probability of mitting a type Ⅰ（分子自由度為2，分母自由度為n3）做比較...2 F and R2l This equation show how F and R^2 are related. These two statistics vary directly, when R^2=0(. no relationship between Y and the X variables), F is zero ipso facto. 兩個變量同方向變動，當R^2=0（即Y與解釋變量X不相關）時，F(xiàn)為0l The larger R^2 is , the greater the F value will be.l In the limit when R^2=1, the F value if infinite. R^2取極限值1時，F(xiàn)值趨于無窮大l Thus the F test discussed earlier, which is a measure of the overall significance of the estimatedregression line, is also a test of significance of R^2, that is, whether R^2 is different from zero.因此，F(xiàn)檢驗也可用于檢驗R^2的顯著性——R^2是否顯著不為零。l In other words, testing the null hypothesis all slope coefficients are equal to zero is equivalentto testing the null hypothesis that R^2 is zero. 檢驗零假設式與檢驗零假設（總體的）R^2為零是等價的2 Specification error設定誤差In our multiple regression that both the age of the clock and the number of bidders variables were individually as well as collectively important influences on the auction price.鐘表年代和競標人數(shù)無論是單獨地還是聯(lián)合地都對拍賣價格有重要影響2 Comparing two R^2 values: the adjusted R^2比較兩個R^2值：校正的判定系數(shù)By examining the R^2 values of these twovariable model and the threevariable model, we can find out that the R^2 value is for agemodel , for the biddersmodel, both are smaller than that of the threevariable model (). 檢查雙變量回歸模型與三變量回歸模型的R^2值，前一個方程的R^2值（）比后面一個方程的R^2值（0,8906）小得多2 The features of the adjusted R^2ba 校正的判定系數(shù) 性質：1. if k1, R^2 bar ≤R^2, that is , as the number of explanatory variables increases in a model, the adjusted R^2ba bee increasingly smaller than the unadjusted R^2. There seem to be a “penalty” involved in adding more explanatory variables to a regression model.隨著模型中解釋變量個數(shù)的增加，校正判定系數(shù)R^2bar越來越小于未校正判定系數(shù)R^2,這似乎是增加解釋變量的“懲罰”although the unadjusted R2 is always positive, the adjusted R2 can on occasion turn out to be is due to its special formula form.雖然未校正判定系數(shù)R^2總為正，但校正判定系數(shù)R^2bar可能為負2 When does adjusted R2 increase ?什么時候增加新的解釋變量R2 bar will increase if the |t| (absolute t) value of the coefficient of the added variable is larger than 1, where the t value is puted under the null hypothesis that the population value of the said coefficient is zero. 如果增加變量系數(shù)的 |t| 值大于1，R^2bar就會增加，這里的t值是在零假設“真實系數(shù)為零”下計算得到的2 Some interesting factsl If you square the t value of , we get ()^2=, which is about the same as theF value of shown before. 如果將t值平方，與F值幾乎相等l It is not surprising because 2 The null hypothesis here is that the restrictions imposed by the restricted model are valid.檢驗的零假設為：受限模型的約束是有效的 If the F value estimated from its statistic exceeds the critical F value at the chosen level of signifi