The article “Analysis of the Modeling Methodologies for Predicting the Strength of Air-Jet Spun Yarns” (Textile Res. J., 1997: 39–44) reported on a study carried out to relate yarn tenacity (y, ∈ gtex) to yarn count (x1, ∈ tex), percentage polyester (x2), first nozzle pessure (x3, ∈ kgcm2), and second nozzle pressure (x4, ∈ kgcm2) The estimate of the constant term in the corresponding multiple regression equation was 6.121. The estimated coefficients for the four predictors were -0.082, 0.113, 0.256, and -0.219, 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. b) Again using n=25, calculate the value of adjusted R2. c) Calculate a 99% confidence interval for true mean yarn tenacity when yarn count is 16.5, yarn contains 50% polyester, first nozzle pressure is 3, and second nozzle pressure is 5 if the estimated standard deviation of predicted tenacity under these circumstances is 0.350.

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The article “Analysis of the Modeling Methodologies for Predicting the Strength of Air-Jet Spun Yarns” (Textile Res. J., 1997: 39–44) reported on a study carried out to relate yarn tenacity (y, ∈ gtex) to yarn count (x1, ∈ tex), percentage polyester (x2), first nozzle pessure (x3, ∈ kgcm2), and second nozzle pressure (x4, ∈ kgcm2) The estimate of the constant term in the corresponding multiple regression equation was 6.121. The estimated coefficients for the four predictors were -0.082, 0.113, 0.256, and -0.219, 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.  b)  Again using n=25, calculate the value of adjusted R2.  c) Calculate a 99% confidence interval for true mean yarn tenacity when yarn count is 16.5, yarn contains 50% polyester, first nozzle pressure is 3, and second nozzle pressure is 5 if the estimated standard deviation of predicted tenacity under these circumstances is 0.350.

izboknil3 · Accepted Answer

Step 1 a) A model utility test for testing H0:β1=β2=⋯=βk=0 versus alternative Ha:at least one, βi≠0, (i=1, 2, ⋯, k) use test statistic value f=R2k(1−R2)[n−(k+1)]=SSRkSSE[n−(k+1)]=MSRMSE The test is upper-tailed - the P-value is the area under Fk, n−(k+1) to the right of the test statistic value. A model utility test is of use. The coefficient of multiple determination, n, and k are given; thus, the value of f statistic is R2k(1−R2)[n−(k+1)]=0.94641−0.94620=87.6 As usual, one could use P-value under the mentioned curve, which is P=0.000, or the table in the appendix, for which the closest values is for α=0.001 F0.001, 4, 20=7.1 Since P=0.00&amp;lt;α reject null hypothesis at reasonable significance level. Similarly same conclusion can be made from the fact F0.001, 4, 20=7.1≤87.6=f Step 2 b) Adjusted R2 is given by 1−n−1n−(k+1)SSESST Using this, and the fact that SSESST=1−R2 the adjusted R2 is 1−n−1n−(k+1)(1−R2)=1−2420(1−0.946)=0.935 c) A 100(1−α)% confidence interval for mean of Y when x=x⋅, μY⋅x⋅, is y^±tα2, n−(k+1)⋅sY For x⋅=(16.5, 50, 3, 5), putting values into the given regression model (estimated coefficients are given)0 the following holds

The article “Analysis of the Modeling Methodologies for Predicting the Strength

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