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, in g/tex) to yarn count (x₁, in tex), percentage polyester (x₂), first nozzle pressure (x₃, in kg/cm²), and second nozzle pressure (x₄, in kg/cm²). The estimate of the constant term in the corresponding multiple regression equation was 6.121. The estimated coefficients for the four predictors were .082, .113, .256, and .219, respectively, and the coefficient of multiple determination was .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 .350.