Concept explainers
Suppose you fit the model
to n = 25 data points with the following results:
- a. Is there sufficient evidence to conclude that at least one of the parameters β1, β2, β3, or β4 is nonzero? Test using α=.05.
- b. Test H0: β1 = 0 against Ha: β1 < 0. Use α =.05.
- c. Test H0: β2 = 0 against Ha: β2 > 0. Use α =.05.
- d. Test H0: β3 = 0 against Ha: β3 ≠ 0. Use α =.05.
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Check out a sample textbook solutionChapter 12 Solutions
Statistics for Business and Economics (13th Edition)
- Given the “data” determined by y = x^3 + (x-1)^2 with x = 0.1, 1.2, 2.3, and 2.9, calculate SSTO, SSR, and R^2. Then recalculate these using x = 0.3, 1, 2.45, and 2.8. Does where you collect your data (i.e., which values of x) appear to impact your interpretation of how good the linear model fits?.arrow_forwardConsider a simple linear regression model with one predictor variable X and one response variable Y. The regression equation is given by Y = β0 + β1X, where β0 and β1 are the intercept and slope coefficients, respectively. The sample size is n = 100 and the coefficient of determination (R^2) for the model is 0.25. What can we conclude about the strength of the relationship between X and Y?arrow_forwardIt is argued that less time spent on social media will result in improved course marks among ECO242 students. To test whether this is the case you collect data from 20 students on their final marks (Y) and number of Instagram posts during the semester (X). You make the following calculations: ∑XY=32191∑X2=34282∑X=690∑Y=1164 Next, you run the following regression: marks=β^1+β^2Instagram where 'marks' is the final mark for the course in percentage points, and 'Instagram' is the average number of minutes per day spent on the Instagram App during the semester. Answer the remaining questions based on your results. Question 1 The value for the slope parameter is? Question 2 CThe value for the intercept parameter is? Question 3 .If the standard error for the intercept parameter estimate is 1.216716, construct a 95% confidence interval for the parameter. Pr( ? ≤β1≤ ?. )=95% Question 4 .If the standard error for the slope parameter…arrow_forward
- For the following data. i_xi_fi 00 1 13 4 267 39 9 4 12 15 i) ii) if data points at i=1,2,3 are considered, write the followings: -the first differences=? -the second differences=? -P₂(s)=? in terms of s, and the first and second differences. -S=? df(x=6)/dx=? By using only two data points.arrow_forwardSuppose that you wish to estimate the effect of class attendance on student performance. A basic model is examscore = β0 + β1attendance + β2priorGP A + u where examscore is students’ score on the exam (from 1 to 6), attendance is the number of TA sessions attended on Zoom (from 0 to 9), and priorGPA is the average exam grade last year. (a) Let internet be the quality of internet at the student’s study place. Do you think internet satisfies the independence assumption? What about the exclusion restriction? (b) Assuming that internet satisfies the conditions above, what other condition must internet satisfy in order to be a valid IV for attendance? (c) Suppose, we add the interaction term priorGP A × attendance. Interpret the coefficient on the interaction term. (d) (Difficult) If attendance is endogenous, then, in general, so is priorGP A × attendance. What might be a good IV for priorGP A × attendance?arrow_forwardFor observed data y=(y1,…,yn)y=(y1,…,yn) with n=21n=21, the above linear regression model was fitted in R, with the following output: >n = 21 >xi = seq(0, n-1,1)/(n-1) >p1 =2*xi-1 >p2 =6*xi^2- 6*xi+1-1/(n-1) > summary(lm(y ~ p1+p2)) Call: lm(formula = y ~ p1 + p2) Residuals: Min 1Q Median 3Q Max -0.5258 -0.2153 0.0813 0.1770 0.4669 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.004238 0.063450 -0.067 0.947 p1 1.181260 0.104784 11.273 1.37e-09 *** p2 -0.953388 0.129422 -7.366 7.77e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2908 on 18 degrees of freedom Multiple R-squared: 0.9097, Adjusted R-squared: 0.8997 F-statistic: 90.68 on 2 and 18 DF, p-value: 3.989e-10 Write this linear regression model in the vector form and answer the following questions, using the above R output where necessary.arrow_forward
- It is argued that less time spent on social media will result in improved course marks among ECO242 students. To test whether this is the case you collect data from 20 students on their final marks (Y) and number of facebook posts during the semester (X). You make the following calculations: ΣXY = 9057; ΣX2 = 2470; ΣX = 190; ΣY = 1164 Next, you run the following regression: marks=β^1+β^2facebookposts; where β^1 = 86.7855 and β^2 = -3.0090 Question: If the standard error for the intercept parameter estimate is 1.43701, construct a 95% confidence interval for the parameter. Pr( _ ≤β1≤ _)=95%arrow_forwardShow that if β^1 is conditionally unbiased, then it is unbiased; that is, showthat if E(β^1|X1, . . . . , Xn) = β1, then E(β^1) = β1.arrow_forwardX” denote the number of children ever born to a woman, and let “Y” denote years ofeducation for the woman. A simple model relating fertility to years of education is X = β0 + β1Y + u where u is the unobserved error. (i) What kind of factors are contained in u? Are these likely to be correlated with level of education?arrow_forward
- Juliet is studying the change in the amount of chemical (in grams) present in her solution as time (in minutes) goes on. She finds a least squares regression line of y = 30 – 2.1x and a coefficient of determination „2 = 0.49. • After 10 minutes, we predict she has grams left. • The correlation coefficient isarrow_forwardConsider the following regression Yi = βXi + ui. Show that (a) the OLS estimator of β is βˆ = (Σ(XiYi))/(Σ(Xi^2)) (b) (Σ(uˆi)/)n = 1/n(Σ(Yi − Yˆi))=/0arrow_forwardA regression and correlation analysis resulted in the following information regarding a dependent variable (y) and an independent variable (x). Σx = 90 Σ(y - )(x - ) = 466 Σy = 170 Σ(x - )2 = 234 n = 10 Σ(y - )2 = 1434 SSE = 505.98 The least squares estimate of the slope or b1 equals a. .923. b. 1.991. c. -1.991. d. -.923.arrow_forward
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