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 $(y,\text{}\in \text{}\frac{g}{tex})$ to yarn count $({x}_{1},\text{}\in \text{}tex)$, percentage polyester $\left({x}_{2}\right)$, first nozzle pessure $({x}_{3},\text{}\in \text{}\frac{kg}{c{m}^{2}})$, and second nozzle pressure $({x}_{4},\text{}\in \text{}\frac{kg}{c{m}^{2}})$ 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 $R}^{2$.

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.