Suppose you fit the first-order multiple-regression model y = b0 + b1 x1 + b2 x2 + e to n = 25 data points and obtain the prediction equation y n = 6.4 + 3.1x1 + .92x2 The estimated standard deviations of the sampling distributions of b n1 and b n2 are 2.3 and .27, respectively.a. Test H0: b1 = 0 against Ha: b1 7 0. Use a = .05. b. Test H0: b2 = 0 against Ha: b2 0. Use a = .05. c. Find a 90% confidence interval for b1. Interpret the interval. d. Find a 99% confidence interval for b2. Interpret the interval.
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Suppose you fit the first-order multiple regression model y = ̠₀ + ̠₁x₁ + ̠₂x₂ + ̵ to n = 25 data points and obtain the prediction equation ŷ = -36.71 + 4.42x₁ + 2.44x₂. The estimated standard deviations of the sampling distributions of ̠₁ and ̠₂ are 0.67 and 0.45, respectively. a. Test H₀: ̠₁ = 0 against Hₐ: ̠₁ > 0. Use ̠ = 0.01. b. Find a 95% confidence interval for ̠₁. Interpret the interval. a. The test statistic is 6.597. (Round to three decimal places as needed.) The p-value is 0.000. (Round to three decimal places as needed.) Reject the null hypothesis. There is sufficient evidence to support the alternative hypothesis. b. What is the confidence interval? ( , ) (Round to three decimal places as needed.)
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A random sample (Yi, Xi) of size n = 250 is drawn and the OLS estimation yields the following (with the sample standard deviation estimated in parentheses): Y = 5.4 + 3.2X (3.1) (1.5) (a) Test H0: β1 = 0 vs. H1: β1 ≠ 0 at the 5% level. (b) Construct a 95% confidence interval for β given the estimate β. (c) Suppose you learned that Yi and Xi were independent. Is it ever possible? Explain. (d) Suppose Yi and Xi are independent and many samples of size n = 250 are drawn, regressions estimated, and the earlier hypotheses are tested and confidence intervals constructed. In what fraction of the samples would H0 from the earlier hypothesis be rejected? And in what fraction of samples would the value 0 be included in the confidence interval for β1?
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