6. (a) Show that in the simple linear regression setting, the usual F test statistic of Họ: B1 = 0 can be written as E- R°(n – 2) 1– R° (b) In a simple linear regression with n = 300, how large must Rề be in order for there to be significant evidence against the null hypothesis from a), at the 5% significance level.
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- A sample of n = 25 pairs of scores (X and Y values) produces a correlation of r = –0.40. Are these sample data sufficient to conclude that there is a significant non-zero correlation between X and Y in the population? Use a two-tailed test at the α = 0.05 level of significance. Which proportion of the variance for the Y scores is predicted by the regression equation?Suppose we have fit a multiple linear regression with 8 explanatory variables and an intercept with 85 observations. We want to test the joint significance of the first 5 explanatory variables using an F test. Please fill in the blanks for the numerator and denominator degrees of freedom of the F statistic of the test: "The F statistic is F(Suppose we perform a simple linear regression (SLR) of two variables F1 and F2 against the same out variable (say y). The r-squared value/score of both the SLRS are 0.9678 and 0.95123 respectively for F1 and F2. Then what can you say about the correlation? O a. y is more dependent on F2 than F1 O b. None of these O . The result does not say that y is more dependent on F1 OR F2 O d.y is more dependent on F1 than F2
- You have estimated a multiple regression model with 6 explanatory variables and an intercept from a sample with 46 observations. What is the critical value of the test statistic (tc) if you want to perform a test for the significance of a single right-hand side (explanatory) variable at α = 0.05? a.) 2.023 b.) 2.708 c.) 2.423 d.) 2.704True or false (Explain why.): A school administrator is running a regression of fall quarter final grades (Y ) on undergraduate grades (X). Since he only has data on graduate students, he does not have people who performed poorly in their undergraduate studies. His sample is therefore nonrandom (selection on X) and her estimates will be biased. True or false (Explain why): Because the IV estimator uses only a subset of the variation in an explanatory variable, precision always increases when compared to OLS.A regression was run to determine if there is a relationship between the happiness index (y) and life expectancy in years of a given country (x). The results of the regression were: ŷ=a+bx a=-0.523 b=0.177 (a) Write the equation of the Least Squares Regression line of the form j= 0.177 +| -0.523 (b) Which is a possible value for the correlation coefficient, r? O 1.936 O -0.632 O-1.936 O 0.632 (C) If a country increases its life expectancy, the happiness index will O decrease O increase (d) If the life expectancy is increased by 1 years in a certain country, how much will the happiness index change? Round to two decimal places. (e) Use the regression line to predict the happiness index of a country with a life expectancy of 64 years. Round to two decimal places.
- If the linear correlation coefficient between the explanatory variable (x) and response variable (y) is r = 0.73, the slope of the regression line is negative O not enought information to answer O positiveCoastal State University is conducting a study regarding the possible relationship between the cumulative grade point average and the annual income of its recent graduates. A random sample of 147 Coastal State graduates from the last five years was selected, and it was found that the least-squares regression equation relating cumulative grade point average (denoted by x, on a 4-point scale) and annual income (denoted by y, in thousands of dollars) was y = 37.79+5.51x. The standard error of the slope of this least-squares regression line was approximately 2.10. Test for a significant linear relationship between grade point average and annual income for the recent graduates of Coastal State by doing a hypothesis test regarding the population slope B1. (Assume that the variable y follows a normal distribution for each value of x and that the other regression assumptions are satisfied.) Use the 0.05 level of significance, and perform a two-tailed test. Then complete the parts below. (If…lecturer would like to know if the foundation Cumulative Grade Point Average (CGPA) predicts the undergraduate CGPA of engineering students at year one. She obtained the relevant information from 105 randomly selected students as follows: Σy = 333.15, Σxy = 1043.626, = 1021.487, > y² = 1077.836 %| Σ %3D (i) Obtain the least square regression line, ŷ = a + bx for the study. (ii) Estimate the strength of the association between the foundation CGPA and CGPA in year one. Interpret the result.
- Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 11 members of the population, where for each member, the value of Y as well as the values of the three explanatory unknowns were observed. The data is entered into a computer using linear regression software and the output summary tells us that R-square is 0.72, the linear model coefficient of the first explanatory unknown is 7 with standard error estimate 2.5, the coefficient for the second explanatory unknown is 11 with standard error 2, and the coefficient for the third explanatory unknown is 15 with standard error 4. The regression intercept is reported as 28. The sum of squares in regression (SSR) is reported as 72000 and the sum of squared errors (SSE) is 28000. From this information, what is MSE/MST? (a) .4000 (b) .3000 (c) .5000 (d) .2000 (e) NONE OF THE OTHERSSuppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 16 members of the population, where for each member, the value of Y as well as the values of the three explanatory unknowns were observed. The data is entered into a computer using linear regression software and the output summary tells us that R-square is 45/62, the linear model coefficient of the first explanatory unknown is 7 with standard error estimate 2.5, the coefficient for the second explanatory unknown is 11 with standard error 2, and the coefficient for the third explanatory unknown is 15 with standard error 4. The regression intercept is reported as 28. The sum of squares in regression (SSR) is reported as 90000 and the sum of squared errors (SSE) is 34000. From this information, what is the critical value needed to calculate the margin of error for a 95 percent confidence interval for one of the model coefficients? (a) 2.069 (b) 2.110 (c)…A sample of n = 15 pairs of X and Y scores produces a Pearson correlation of r = 0.45, SSY = 90. a) If the regression equation was found for these scores, how much of the Y variability would be predicted by the regression equation (SSregression) and how much would not be predicted (SSresidual)? b) Does the regression equation predict a significant portion of the variability for the Y scores? (Equivalently, is the Pearson correlation significant?)