DSC5100 HW2 (1)

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Apr 3, 2024

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1) Fitted model: y i = 0.830 - 0.818x i + ɛ i 2) x = 2.107 y = 0.830 - 0.818 * 2.107 y ≈ -0.665 Predicted value of y when x = 2.107 is -0.665. 3) The intercept of the least-squares regression line is 0.830. 4) H0: β 1 = 0, t = (-0.818 - 0) / 0.313 t ≈ -2.615 The t statistic is -2.615. 5) For H0: β 0 = 0, t = (0.830 - 0) / 0.275 t ≈ 3.018 The t statistic is approximately 3.018. 6) The 95% confidence interval for the slope β 1 Given a significant level of 0.05 and n = 55, the critical value is about 2.004. The confidence interval for β 1 is: -0.818 ± (2.004*0.313) -0.818 ± 0.627 Thus, the 95% confidence interval for the slope β 1 is approximately (-1.445, -0.191). 7) The 95% confidence interval for the intercept β 0 is: 0.830 ± (2.004*0.275) 0.830 ± 0.551 So, the 95% confidence interval for the intercept β 0 is approximately (0.279, 1.381). 8) The correlation coefficient r between x and y is:
r = √(R 2 ) R 2 = 0.675 r = √(0.675) r ≈ 0.821 So, the correlation coefficient r between x and y is about 0.821. 9) R 2 represents the proportion of the total variation in the dependent variable that can be explained by the independent variable in the model. In other words, it indicates how well the independent variables can predict or explain the variation observed in the dependent variable. So, when R 2 = 0.675, that means that 67.5% of the variation in the dependent variable y can be attributed to the independent variable x . 10) In a simple linear regression model, the intercept represents the predicted value of the dependent variable y, when the independent variable is 0. So, when the intercept value is .830, the independent variable is 0. In other words, if x is 0, y is predicted to be .830. 11) SST (total sum of squares) represents the total variation within the dependent variable y around its mean. In order to calculate SST, we sum the squared differences between each observed y value and the mean of y . SST = Σ(y i - ȳ)² 12) In order to assess the significance of both the intercept and slope for the analysis, we compare their calculated t-statistic values to critical t-values at a 5% significance level. If the calculated t statistic is more than the critical t-value (in absolute terms), we reject the null hypothesis and assume that there is statistical significance. For the slope (β 1 ) hypothesis check: Calculated t statistic = -2.615 degrees of freedom: = n – 2, 55- 2 = 53 Significance level = 0.5 Critical t value = ± 2.004 -2.615 < 2.004 For the intercept (β 0 ) hypothesis: t statistic = 3.018 Significance level = 0.5
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