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

Lipossig

Lipossig

Answered question

2021-09-29

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.

Answer & Explanation

izboknil3

izboknil3

Skilled2021-09-30Added 99 answers

Step 1
a) A model utility test for testing
H0:β1=β2==βk=0
versus alternative
Ha:at least one, βi0, (i=1, 2, , k)
use test statistic value
f=R2k(1R2)[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(1R2)[n(k+1)]=0.946410.94620=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<α
reject null hypothesis at reasonable significance level. Similarly same conclusion can be made from the fact
F0.001, 4, 20=7.187.6=f
Step 2
b) Adjusted R2 is given by
1n1n(k+1)SSESST
Using this, and the fact that
SSESST=1R2
the adjusted R2 is
1n1n(k+1)(1R2)=12420(10.946)=0.935
c) A 100(1α)% confidence interval for mean of Y when x=x, μYx, 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

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