# Question

Use the results of Exercises 14.20 and 14.21 and the fact that E(Bˆ) = β and var(Bˆ) = σ2/ Sxx to show that Y0 – (Aˆ + Bˆx0) is a random variable having a normal distribution with zero mean and the variance

Here Y0 has a normal distribution with the mean α + βx0 and the variance σ2; that is, Y0 is a future observation of Y corresponding to x = x0. Also, use the first part of Theorem 14.3 as well as the fact that Y0 – (Aˆ + Bˆx0)and n∑ˆ2/σ2 are independent to show that

Is a value of a random variable having the t distribution with n – 2 degrees of freedom.

Here Y0 has a normal distribution with the mean α + βx0 and the variance σ2; that is, Y0 is a future observation of Y corresponding to x = x0. Also, use the first part of Theorem 14.3 as well as the fact that Y0 – (Aˆ + Bˆx0)and n∑ˆ2/σ2 are independent to show that

Is a value of a random variable having the t distribution with n – 2 degrees of freedom.

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