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.
Answer to relevant QuestionsSolve the double inequality –tα/2,n–2 < t < tα/2,n–2 with t given by the formula of Exercise 14.25 so that the middle term is y0 and the two limits can be calculated without knowledge of y0. Although the resulting ...By solving the double inequality –zα/2 ≤ z ≤ zα/2 (with z given by the formula on page 402) for ρ, derive a (1 – α) 100% confidence interval formula for ρ. On page With x01, x02, . . . , x0k and X0 as defined in Exercise 14.39 and Y0 being a random variable that has a normal distribution with the mean β0 + β1x01 + · · · + βkx0k and the variance σ2, it can be shown that Is a ...During its first five years of operation, a company’s gross income from sales was 1.4, 2.1, 2.6, 3.5, and 3.7 million dollars. Use the coding of Exercise 14.15 to fit a least squares line and, assuming that the same linear ...With reference to Example 14.4, use the theory of Exercise 14.22 to test the null hypothesis α = 21.50 against the alternative hypothesis α ≠ 21.50 at the 0.01 level of significance. In exercise Use the result of ...
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