A study was carried out to relate yarn tenacity (y, in g/tex) to yarn count (x_1
ID: 3131439 • Letter: A
Question
A study was carried out to relate yarn tenacity (y, in g/tex) to yarn count (x_1, in tex), percentage polyester (x_2), first nozzle pressure (X_3, in kg/cm^2), and second nozzle pressure (x_4, in kg/cm^2). The estimate of the constant term in the corresponding multiple regression equation was 6.128. The estimated coefficients for the four predictors were -0.082, 0.111, 0.258, and -0.216, respectively, and the coefficient of multiple determination was 0.946. (a) Assuming that the sample size was n = 25, state and test the appropriate hypotheses to decide whether the fitted model specifies a useful linear relationship between the dependent variable and at least one of the four model predictors. (Use a = 0.001.) State the appropriate hypotheses. State the rejection region and compute the test statistic value. Round your answers to two decimal places. State the conclusion in the problem context. Fail to reject H_0. The model is judged not useful. Fail to reject Hq. The model is judged useful. Reject H_0. The model is judged not useful. Reject H_0. The model is judged useful. Again using n = 25, calculate the value of adjusted R^2. (Round your answer to three decimal places. Read section on page 546 in the text.) (c) Calculate a 99% confidence interval for true mean yarn tenacity when yarn count is 16.5, yarn contains 60% polyester, first nozzle pressure is 2, and second nozzle pressure is 5 if the estimated standard deviation of predicted tenacity under these circumstances is 0.34. ( df = n-(k+l). Polyesters unit is percent, ex. 50% = 50 not 0.50.)Explanation / Answer
Here we want ot test whether any of these independent variables has a significant effect or not. The measure of significance is tested by the nullity of the respective regression coefficients. So our regression will be significant if atlest one of the coefficients is non-zero.
So our test of hypothesis will be
H0 : b1=b2=b3=b4=0 vs H1 : atlest one of them is non-zero.
So option (a) will be correct.
you have not provided the regression table and the values of independent variables. With only the betas one can't find the value of the F-statistic. Now if the value of the value of the F-statistic is greater than 7.10 then we will reject null hypothesis in favour of alternative and conclude that atleast one of the predictors has a significant effect on the dependent variable. If it is less than 7.10 then will accpet null and say that the predictor have no significant effect on the dependent variable. If it is greater than 7 then the answer is (4) and if less than the answer is (2).
(b) we know that coefficient of multiple determination (R) is 0.946. So the adjusted R-square will be
= 1- (1-R2)*(n-1)/(n-p-1) where n=25 and p=number of predictors= 4.
So adjusted R-square= 0.874
(c) expected value of the yarn tenacity when X1= 16.5, X2=60, X3=2, X4=5 is
= 6.128 + (-0.082)*16.5 +0.111 * 60 + 0.258 * 2 + (-0.216)*5 = 10.871 g/tex.
So a 99% confidence interval is (10.871 - t0.005,(25-4-1) * 0.34 , 10.871 + t0.005,(25-4-1) * 0.34 ) ,
where t0.005,(25-4-1) is the upper 0.5% point of t distribution with (25-1-4)=20 degress of freedom.
Now t0.005,(25-4-1)= 2.845, So the 99% confidence interval is given by
(9.904,11.838) .
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