Description

(a) Using children who are in either a regular-sized class or a small class, estimate theregression model explaining students’ combined aptitude scores as a function of class size, TOTALSCOREi = β1 + β2SMALL+ei. Interpret the estimates. Based on this regression result, what do you conclude about the effect of class size on learning?(Hint:Remember to exclude classes with teacher aide from the sample)
(b) Repeat part (a) using dependent variables READSCORE and MATHSCORE. Do you observe any differences?
(c) Using children who are in either a regular-sized class or a regular-sized class with ateacher aide, estimate the regression model explaining student’s combined aptitude scores as a function of the presence of a teacher aide,TOTALSCOREi = γ1+γ2AIDE+ ei. Interpret the estimates. Based on this regression result, what do you conclude about the effect on learning of adding a teacher aide to the classroom? (Hint:Remember to exclude small classes from the sample)
(d) Repeat part (c) using dependent variables READSCORE and MATHSCORE. Do you observe any differences?
2. You have the results of a simple linear regression based on state-level data and the Districtof Columbia, a total of N = 51 observations.
(a) The estimated error variance ˆσ2=2.04672. What is the sum of the squared least squares residuals?
(b) The estimated variance of b2 is 0.00098. What is the standard error of b2? What is the value of P(xi − x¯)2?
(d) Suppose ¯x=69.139 and ¯y= 15.187, what is the estimate of the intercept parameter?
(e) Given the results in (b) and (d), what is Px2i ?
(f) For the state of Arkansas the value of yi= 12.274 and the value of xi=58.3. Compute the least squares residual for Arkansas.(Hint:Use the information in parts (c) and (d))
3. The general manager of an engineering firm wants to know whether a technical artist’s experience influences the quality of his or her work. A random sample of 24 artists is selected and their years of work experience and quality rating (as assessed by their supervisors) recorded. Work experience (EXPER) is measured in years and quality rating (RATING) takes a value of 1 through 7, with 7 = excellent and 1 = poor: The simple regression model RATING = β1 +β2EXPER+e is proposed. The least squares estimates of the model, and the standard errors of the estimates, are RATINGd = 3.204 + 0.076EXPER
(se) (0.709) (0.044)
(a) Sketch the estimated regression function. Interpret the coefficient of EXPER.
(b) Construct a 95 % confidence interval for β2, the slope of the relationship between quality rating and experience. In what are you 95 % confident?
(c) Test the null hypothesis that β2 is zero against the alternative that it is not using a two-tail test and the α = 0.05 level of significance. What do you conclude?
(d) Test the null hypothesis that β2 is zero against the one-tail alternative that it is positive at the α = 0.05 level of significance. What do you conclude?
(e) For the test in part (c), the p-value is 0.0982. If we choose the probability of a Type I error to be α = 0.05; do we reject the null hypothesis, or not, just based on an inspection of the p-value?