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The Distribution of F. The distribution of F depends on the shape of the distribution of the random errors. For testing hypotheses by the method of least-squares, we assume that the errors are normally distributed. The resulting distribution of F, when the null hypothesis is true, is the F distribution with 4 and n - 5 degrees of freedom. The Turnip Green Data. From the data in Table 1.1 we can calculate SSR ful1 = 819.7, (j2 = 819.7/22 = 37.26, and SSRrcduccd = 9150.5. So F = (9150.5 - 819.7)/4(37.26) = 55.90. The degrees of freedom are 4 and 22. The calculation of SSR full is very difficult to do with an ordinary hand calculator, and so one would want to have the calculation of the test statistic done by a regression computer package or else write a computer program oneself (see Section 3.12). Most regression computer packages would print out the p-value of the test. Some computer languages include a function that would yield the p-value given the input 55.90, 4, and 22. The p-value is less than 0.0001. Without a computer, one can pin down the p-value by using tables called F tables. The F table in the Appendix tells us that for 4 and 22 degrees of freedom, Prob(F z 4.31] = 0.01 when the null hypothesis is true. Since 55.90 > 4.31, we see that the p-value of our test is less than 0.01. So we can conclude that vitamin B2 concentration in turnip greens has a significant linear relationship with at least some of the four variables sunlight, soil moisture, air temperature, and squared soil moisture.

void generateCrossReference( private: Tokenizer tok;

(2'17 )

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The xref class interface is shown in Figure 1328 It is similar to (but simpler than) the Balance class shown in Figure 123, which was part of a balanced symbol program Like that class, it makes use of the Tokenizer class defined in Figure 122 We can now discuss the implementation of the two remaining routines in the ~ o k e n i z e r class: g e t ~ e x and g e t ~ e m a i n i n g ~ t r i n g t ~ ~ These new parsing routines deal with recognizing an identifier A C++ identifier consists of alphanumeric characters and underscores with the restriction that the first character may not be a digit Consequently, the routine shown in Figure 1329 tests whether a character is part of an identifier In the getRemainingString routine shown in Figure 1330 we assume that the first character of an identifier has already been read and is stored in the Tokenizer class data member ch It repeatedly reads charac-

1 / / Return indicates if ch can be part of a C++ identifier 2 boo1 isIdChar( char ch ) 3 { 4 return ch == ' - ' / 1 isalnum( ch ) ; 5 }

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

(12.23)

f33 = 0

string Tokenizer::getRemainingString( )

string result;

Thus, if from the observed positron spectrum we plot Pe- 1(df /dpj/2 as a function of E e, we should obtain a linear plot with end point Eo. This is called the Kurie plot. It can be used to check whether the neutrino mass is indeed zero. A nonvanishing neutrino mass destroys the linear behavior, particularly for E e near Eo. (In practice, of course, we must examine the approximations we have made, correct E e for the energy gained from the nuclear Coulomb field, and allow for the experimental energy resolution.) Our immediate interest here is of a different nature. We wish to determine G from the observed value Eo and the measured lifetime 'T of the nuclear state to /3-decay. We therefore carry out the dPe integration of (12.23) over the interval 0, Eo. Making the relativistic approximation Pe == E e, we find

f o r ( result = ch; nextchar( ) ;

3.9 TESTING f3 q +I = ... = f3 p = 0 Consider the general form of the linear regression model, Y = f30 + f3IXI + ... +f3pXp + e. The testing procedure described in the preceding section can be extended to allow comparison of this model with the reduced model formed from any subset of the explanatory variables. By changing indices, we can write the reduced model as Y = f3 0 + f3IXI + ... +f3qXq, where q < p. Note that comparing these two models is the same as testing f3 q+ I = ... = f3 p = O. We continue to use the notation SSR full and SSRreduced for the sums of squares of the residuals for the two models. Let 0- 2 denote the following unbiased estimate of (J"2: SSR ful1

) ) ;

putBackChar( break;

12.5 Muon Decay the measured half-life is 'T log 2 = 71 sec. Using this information, we find G == 10- 5/m~.

return result; 1

ters until one that is not part of an identifier appears At that point we put the character back (at line 10) and then return a string The getNextID routine shown in Figure 133 1 is similar to the routine shown in Figure 127 The difference is that here at line 22, if the first character of an identifier is encountered, we call getRemainingString to return the token The fact that getNextID and getNextOpenClose are so similar suggests that it would have been worthwhile to write a private member function that performs their common tasks With all the supporting routines written, let us consider the only member function, generateCrossRef erence, shown in Figure 1332 Line 7 declares the map We read the input and build the map at lines 11 and 12 At each iteration, we have the current identifier Let us see how the expression at line 12 works There are two cases:

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