Confidence intervals questions and answers

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Confidence intervals
asked 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 \(\displaystyle{\left({y},\ \in\ {\frac{{{g}}}{{{t}{e}{x}}}}\right)}\) to yarn count \(\displaystyle{\left({x}_{{{1}}},\ \in\ {t}{e}{x}\right)}\), percentage polyester \(\displaystyle{\left({x}_{{{2}}}\right)}\), first nozzle pessure \(\displaystyle{\left({x}_{{{3}}},\ \in\ {\frac{{{k}{g}}}{{{c}{m}^{{{2}}}}}}\right)}\), and second nozzle pressure \(\displaystyle{\left({x}_{{{4}}},\ \in\ {\frac{{{k}{g}}}{{{c}{m}^{{{2}}}}}}\right)}\) 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 \(\displaystyle{n}={25}\), 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 \(\displaystyle{n}={25}\), calculate the value of adjusted \(\displaystyle{R}^{{{2}}}\).
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.