Suppose you are interested in estimating the effect of hours...

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 particular 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 preparation course, and you can only randomly sample sat and hours from the population. Write the population model as $$\text {sat}=\beta_{0}+\beta_{1} \text {hours}+u$$ where, as usual in a model with an intercept, we can assume $\mathrm{E}(u)=0 .$ 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 $\beta_{1}$ if the preparation course is effective? iv. In the equation from part (ii), what is the interpretation of $\beta_{0} ?$

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 particular 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 preparation course, and you can only randomly sample sat and hours from the population. Write the population model as
$$\text {sat}=\beta_{0}+\beta_{1} \text {hours}+u$$
where, as usual in a model with an intercept, we can assume $\mathrm{E}(u)=0 .$ 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 $\beta_{1}$ if the preparation course is effective?
iv. In the equation from part (ii), what is the interpretation of $\beta_{0} ?$

Key Concepts

Randomized Controlled Experiment

A randomized controlled experiment involves randomly assigning participants to different levels or groups of a treatment—in this case, varying amounts of preparation course hours. Random assignment ensures that, on average, both observed and unobserved characteristics are balanced across groups, which allows for an unbiased estimation of the causal effect of course hours on SAT scores.

Observational Study and Self-Selection

In an observational study where students choose how much time to spend in a preparation course, self-selection bias is a critical issue because the decision to take more or fewer hours of preparation is influenced by factors that might also affect SAT scores. These factors can confound the observed relationship between hours and SAT scores, making it difficult to isolate the causal impact of the course hours.

Omitted Variable Bias

In the context of the population model, omitted variable bias occurs when variables that influence SAT scores—such as a student's innate ability, motivation, socioeconomic status, or quality of schooling—are not included in the model. These unobserved factors, contained in the error term (u), can be correlated with hours spent in the course, leading to biased estimates of the treatment effect.

Regression Coefficient Interpretation

Within the linear model, the coefficient ?1 represents the marginal effect of an additional hour spent in the preparation course on the total SAT score. If the course is effective, ?1 is expected to be positive, indicating that more hours of preparation are associated with higher scores. Meanwhile, ?0, the y-intercept, represents the expected SAT score for a student who does not spend any time in the preparation course.

Transcript

00:01 So suppose that you're interested in estimating the effect of hours spent in the sat preparation costs on total sats ' scores, and population is all college -bound high school senior for a particular year.

00:14 Now, the first question says, suppose you're giving a grant to run a controlled experiment.

00:23 Explain how you would structure the experiment in under to estimate the casual effect of hours on sats.

00:32 So we would want to randomly assign the number of hours in the preparation course so that the hours are independent of other factors that affect the performance on the setc.

00:44 Then you would collect the information on the sats score for each student in the experiment that's using a dataset where n is the number of students we can afford to have in the study.

00:56 So we would want to randomly assign the number of hours in the preparation cost.

01:34 So the hours is independent of other factors that affect performance in the sat.

02:07 After that, you want to collect information on the sat score for each student in the department, in the experiment.

02:31 And that yields a dataset and the dataset is going to look like this with sat, then i is equal to 1 until n where n is the number of students we can afford to have in study so that's it so that's the answer to how you structure the experience and that's estimate the casual effect of hours and sat.

03:22 Now the second question, consider the more or less the more realistic case where students choose how much time they spend in the preparation cost and you can only randomly sample sats and hours from the population.

03:39 And the population model is equal to sats equal to b0 plus b1 hours plus you...

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