Lab 02 - Confidence Intervals
PAF 573
Maryan Robledo
In this assignment you will be working with simulated data on class size and test scores. There is also a variable for socio-economic status and teacher quality. We would like to understand how the relationship between class size and test scores changes when other variables are considered. We will be looking specifically at the confidence interval around the classroom size slope estimate.
We estimate the following models:
\(TestScore = b_0 + b_1 \cdot ClassSize + e_1 \ \ \ (Model \ 1)\)
\(TestScore = b_0 + b_1 \cdot ClassSize + b_2 \cdot TeacherQuality + e_2 \ \ \ (Model \ 2)\)
\(TestScore = b_0 + b_2 \cdot TeacherQuality + b_3 \cdot SES + e_3 \ \ \ (Model \ 3)\)
\(TestScore = b_0 + b_1 \cdot ClassSize + b_3 \cdot SES + e_4 \ \ \ (Model \ 4)\)
\(TestScore = B_0 + B_1 \cdot ClassSize + B_2 \cdot TeacherQuality + B_3 \cdot SES + \epsilon \ \ \ (Model \ 5)\)
| Dependent Variable: Test Scores | |||||
| Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
| (1) | (2) | (3) | (4) | (5) | |
| Classroom Size | -4.22*** | -3.91*** | -2.67 | -2.22*** | |
| (0.18) | (0.03) | (1.63) | (0.23) | ||
| Teacher Quality | 55.01*** | 55.03*** | 55.01*** | ||
| (0.25) | (0.26) | (0.25) | |||
| Socio-Economic Status | 40.94*** | 16.34 | 17.77*** | ||
| (0.27) | (17.10) | (2.40) | |||
| Intercept | 738.34*** | 456.70*** | 272.91*** | 665.29*** | 377.26*** |
| (4.88) | (1.48) | (1.39) | (76.57) | (10.82) | |
| Observations | 1,000 | 1,000 | 1,000 | 1,000 | 1,000 |
| Adjusted R2 | 0.36 | 0.99 | 0.99 | 0.36 | 0.99 |
| Standard errors in parentheses | p<0.1; p<0.05; p<0.01 | ||||
Lab-02 Questions:
Warm-up: Interpret the slope associated with Class Size in Model 01. What does a slope of -4.22 mean in this context? Is the negative sign a good thing or a bad thing?
Answer: The slope of -4.22 means that every additional student in a classroom, the average test scores decrease by around -4.22 points. From an educational standpoint, the negative sign is a bad thing becuase it means that larger class sizes can result in worse student outcomes.
Q (1)
What is the standard error associated with the slope on class size in Model 1?
Answer: From the table, we can see that the standard of error is 0.18.
Q (2)
Calculate the 95% confidence interval around the class size coefficient in Model 1. Is it statistically significant at this level? How do you know?
Answer: The 95% confidence interval around the class coefficient is from -4.57 and -3.87. This is significant because it does not contain zero. -4.22 ± 1.96 (0.18)
Visual
Q (3)
Calculate the 95% confidence interval around the class size coefficient in Model 2. Is it significant at this level? How do you know?
Answer: The 95% confidence interval around the class coefficient is from -3.97 and -3.85. This is significant because it does not contain a zero. -3.9 ± 1.96(0.03)
Visual
Q (4)
Calculate the 95% confidence interval around the class size coefficient in Model 4. Is it significant at this level? How do you know?
Answer: The 95% confidence interval around the class coefficient is from -5.86 and -0.52. This is not significant because it does contain a zero. -2.67 ± 1.96 (1.63)
Visual
Q (5)
Draw the three confidence intervals to see how they change as a result of the controls included in the model.
Which model has the “largest” slope? Note that the slope represents program impact, in this case how much test scores improve for as you reduce average class size by a student. So largest slope, or program effect size, is in absolute terms. A slope of -5, for example, means that test scores improve by 5 points when average class size falls by one student in a state. A slope of -3 means that test scores only improve by 3 points for the one-student reduction in class size. A slope of -3 is larger than -5 in mathematical terms, but when asked about slope size or program effects you should ignore the sign and compare absolute values. The intervention is about reducing class size, so we want a “large” negative slope in this context.
Which model has the smallest standard error? How can you tell?
You can reference the above graphics or re-create them here.
Answer: Model 1 has the largest slope of an absolute value of 4.22. Model 2 has the smallest standard of error of 0.03.
Q (6)
The covariance of class size and test scores is -418, and the variance of class size is 99. Can you calculate the slope of class size in Model 4 with the formula cov(x,y)/var(x)? Why or why not?
Answer: No you would not be able to calculate the slope of class size for model 4. That formula would only work if there was only one independent variable. Model 4 includes both class size and SES. So becuase there is more than one variable the formula wouldn’t work.
Pairs plot for reference for Q6.
