6. (a) Show that in the simple linear regression setting, the usual F test statistic of Ho: ß1 = 0 can be written as R°(n – 2) F = 1– R² (b) In a simple linear regression with n = 300, how large must Rª be in order for there to be significant evidence against the null hypothesis from a), at the 5% significance level.
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- 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(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?True 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 positiveSuppose 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
- 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 OTHERSYou 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.) 2.704. Researchers believed that an increase in lean body mass is associated with an increase in maximal oxygen uptake. A scatterplot of the measurements taken from 18 randomly selected college athletes displayed a strong positive linear relationship between the two variables. A significance test for the null hypothesis that the slope of the regression line is 0 versus the alternative that the slope is greater than 0 yielded a p-value of 0.04. Which statement is an appropriate conclusion for the test? (A) The p-value of 0.04 indicates that 4% of the variation in maximal oxygen uptake for college athletes can be explained by the amount of lean body mass. (B) The p-value of 0.04 indicates that 16% of the variation in maximal oxygen uptake for college athletes can be explained by the amount of lean body mass. (C) The strong positive linear relationship displayed in the scatterplot along with a p-value less than 0.05 indicates that college athletes with higher lean body mass tend to have higher…
- Suppose 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)…Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 12 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 0.85, 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 85000 and the sum of squared errors (SSE) is 15000. From this information, what is SSE/SST? (a) .2 (b) .13 (c) NONE OF THE OTHERS (d) .15 (e) .25Suppose that Y is normal and we have three explanatory unknowns which are also normal, and we have an independent random sample of 21 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 0.9, 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 10000. From this information, what is the number of degrees of freedom for the t-distribution used to compute critical values for hypothesis tests and confidence intervals for the individual model…
