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11 Suppose you are interested in estimating the effect of hours spent in an SAT preparation course (hours) on total SAT score (sat). The population is all college-bound high school seniors for a par- ticular year. (i) Suppose you are given a grant to run a controlled experiment. Explain how you would structure the experiment in order to estimate the causal effect of hours on sat. (ii) Consider the more realistic case where students choose how much time to spend in a prepara- tion course, and you can only randomly sample sat and hours from the population. Write the population model as sat = Bo + B₁hours + u where, as usual in a model with an intercept, we can assume E(u) = 0. List at least two factors contained in u. Are these likely to have positive or negative correlation with hours? (iii) In the equation from part (ii), what should be the sign of B, if the preparation course is effective? (iv) In the equation from part (ii), what is the interpretation of Bo?

11 Suppose you are interested in estimating the effect of hours spent in an SAT preparation course (hours) on total SAT score (sat). The population is all college-bound high school seniors for a par- ticular year. (i) Suppose you are given a grant to run a controlled experiment. Explain how you would structure the experiment in order to estimate the causal effect of hours on sat. (ii) Consider the more realistic case where students choose how much time to spend in a prepara- tion course, and you can only randomly sample sat and hours from the population. Write the population model as sat = Bo + B₁hours + u where, as usual in a model with an intercept, we can assume E(u) = 0. List at least two factors contained in u. Are these likely to have positive or negative correlation with hours? (iii) In the equation from part (ii), what should be the sign of B, if the preparation course is effective? (iv) In the equation from part (ii), what is the interpretation of Bo?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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11 Suppose you are interested in estimating the effect of hours spent in an SAT preparation course (hours) on total SAT score (sat). The population is all college-bound high school seniors for a par- ticular year. Suppose you are given a grant to run a controlled experiment. Explain how you would structure the experiment in order to estimate the causal effect of hours on sat. (ii) Consider the more realistic case where students choose how much time to spend in a prepara- tion course, and you can only randomly sample sat and hours from the population. Write the population model as sat= Bo + B₁hours + u where, as usual in a model with an intercept, we can assume E(u) = = 0. List at least two factors contained in u. Are these likely to have positive or negative correlation with hours? (iii) In the equation from part (ii), what should be the sign of B, if the preparation course is effective? (iv) In the equation from part (ii), what is the interpretation of Bo? 12 Consider the problem described at the end of Section 2-6, running a regression and only estimating an intercept. Given a sample {y: i = 1, 2,..., n}, let Bo be the solution to min Σ(v₁ – bo)². 1 bo i=1 Show that B = y, that is, the sample average minimizes the sum of squared residuals. (Hint: You may use one-variable calculus or you can show the result directly ing y inside the squared residual and then doing a little algebra.) adding and subtract- (ii) Define residuals , = y; - y. Argue that these residuals always sum to zero.
Transcribed Image Text:11 Suppose you are interested in estimating the effect of hours spent in an SAT preparation course (hours) on total SAT score (sat). The population is all college-bound high school seniors for a par- ticular year. Suppose you are given a grant to run a controlled experiment. Explain how you would structure the experiment in order to estimate the causal effect of hours on sat. (ii) Consider the more realistic case where students choose how much time to spend in a prepara- tion course, and you can only randomly sample sat and hours from the population. Write the population model as sat= Bo + B₁hours + u where, as usual in a model with an intercept, we can assume E(u) = = 0. List at least two factors contained in u. Are these likely to have positive or negative correlation with hours? (iii) In the equation from part (ii), what should be the sign of B, if the preparation course is effective? (iv) In the equation from part (ii), what is the interpretation of Bo? 12 Consider the problem described at the end of Section 2-6, running a regression and only estimating an intercept. Given a sample {y: i = 1, 2,..., n}, let Bo be the solution to min Σ(v₁ – bo)². 1 bo i=1 Show that B = y, that is, the sample average minimizes the sum of squared residuals. (Hint: You may use one-variable calculus or you can show the result directly ing y inside the squared residual and then doing a little algebra.) adding and subtract- (ii) Define residuals , = y; - y. Argue that these residuals always sum to zero.
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