Question

Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). x y 2 12 3 9 6 8 7 7 8 6 7 5 9 2 Given the followings b1 = -1.13 SSR = 50.625 SSE= 9.376 Sb1 = .2165 a. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. (No excel please show steps thank you)

Answer 1

Step:1 State the Hypotheses:- If there is a significant linear relationship between the independent variable X and the dependent variable Y, the slope will not equal zero. Ho: Β1 = 0 H1: Β1 ≠ 0 The null hypothesis states that the slope is equal to zero, and the alternative hypothesis states that the slope is not equal to zero. Degrees of freedom. For simple linear regression (one independent and one dependent variable), the degrees of freedom (DF) is equal to: DF = n - 2 where n is the number of observations in the sample. Based on the above table, the following is calculated: Step:2 Test statistic:- The test statistic is a t statistic (t) defined by the following equation. t = b_1-Β1/SE but, Ho: Β1 = 0 Therefore, t = b1 / SE where, b1:The slope of the simple regression line. SE:The standard error of the regression slope. test statistics= t = b1 / SE = -1.125/0.216506 = -5.1962 t_table = 2.5706 we are ready to compute the margin of error : me:Margin of error. me=t_table*se = 2.5706*0.216506 = 0.5566 Then we calculate the confidence interval for regression slope: CI=([b_1-me,b_1+me])=[-1.6816 -0.5684] p_value = 0.0035 Step:3 Decision:- if p_value is less than the significance level 0.05 then we are rejecting the null hypothesis,otherwise we accept it. Step:4 Conclusion:- we are rejecting null hypothesis there is a significant linear relationship between the independent variable X and the dependent variable Y