Quantitative Methods

Tennessee's Project STAR

A longitudinal experiment was conducted in Tennessee beginning in 1985 and ending in 1989. A single cohort of students was followed from kindergarten through third grade. In the experiment children were randomly assigned within schools into three types of classes:

1. small classes with 13-17 students,
2. regular-sized classes with 22-25 students, and
3. regular-sized classes with a full-time teacher aide to assist the teacher.

Student scores on achievement tests for maths and reading were recorded, as was some information about the students, teachers, and school. A subsample of the data on achievement tests' scores is contained in the EViews file star.wf1.

The series in this file are:
math_aide maths score in regular class with a teacher aide math_reg maths score in regular class without a teacher aide math_small maths score in a small class
read_aide reading score in regular class with a teacher aide read_reg reading score in regular class without a teacher aide read_small reading score in a small class
Begin by creating three new series, say total_aide, total_reg, and total_small, which are equal to the sums of the maths and reading scores.
Among other things, we wish to investigate whether (a) having a teacher aide improves total scores, and (b) whether having a small class improves total scores.

Question 1
(a) Find and report the sample means and variances for total_aide and total_reg.
(b) Find the ratio of the two sample variances in (a).
(c) The ratio in (b) can be used to test a null hypothesis. What is that null hypothesis? Be sure to define your notation.
(d) Find the p-value of the test outcome from (b) and (c). Assume a two-tail test.
(e) Assuming a 10% significance level, what is the outcome of the test?
(f) Use EViews to find the exact critical values of the test.
(g) Find a pooled variance estimate under the assumption that the null hypothesis in (b) is true.

Question 2
This question is concerned with testing whether the mean total score for classes with a teacher aide is greater than the mean total score for regular-sized classes without a teacher aide.
(a) Specify suitable null and alternative hypotheses. Be sure to define your notation.
(b) Using information from Question 1 parts (a) and (g), compute the value of the relevant test statistic.
(c) Find the p-value for the value computed in (b).
(d) At a 5% significance level, what is your test conclusion?
(e) Use EViews to find the exact test critical value.

Question 3
This question is concerned with testing whether the mean total score for small-sized classes is greater than the mean total score for regular-sized classes (without a teacher aide).
Use EViews to carry out this test. Include your EViews output. Report your findings.
(There is no need to give step-by-step details. Simply report your findings in two sentences. The first in language you might use for a statistician, and the second in language you might use for the parents of the children in the schools.)

Question 4
Consider the maths scores and reading scores for students in small classes.
(a) Find and report the sample means, variances, and covariance for these two scores.
(b) Find the difference in the sample means and an estimate of the variance of this difference.
(c) Find a 95% interval estimate (confidence interval) for the population difference in the means for maths and reading scores in small classes.
(d) Report the Jarque-Bera p-values for each of the two series. Is it reasonable to assume they come from normal distributions? Comment on the relative shapes of the two histograms. (Include the EViews histograms in your answer.)