Consider normal populations A and B with a common population variance. One sample of size 18 is selected from the normal population A, denoted as , and the other one of size 18 from the normal population B, denoted as . The statistics derived from the samples are as follows.
The values 2.110 and 2.032 may be used for the upper 2.5% point of the -distribution with 17 and 34 degrees of freedom, respectively.
(1) Assuming that the samples are paired as , and randomly selected, compute the 95% confidence interval for the difference between the two population means using necessary statistics among those mentioned above.
(2) Assuming that the samples are unpaired and randomly selected from each population, compute the 95% confidence interval for the difference between the two population means using necessary statistics among those mentioned above.
Consider a linear regression model where the variation in random variables are explained by the corresponding constants with the regression coefficients of and . Assume that are independent and follow a normal distribution with mean 0 and variance . Let and be the least squares estimators of and , respectively.
(1) Compute the standard deviation of .
(2) Given a new constant , show the 95% prediction interval of using , and where is independent of and follows a normal distribution . The value 2.145 for the upper 2.5% point of the -distribution with 14 degrees of freedom may be used.