# Hypothesis Tests & Confidence Intervals

In the 1980s, Tennessee conducted an experiment in which kindergarten students were randomly assigned to "regular" and "small" classes, and given standardized tests at the end of the year. ("Regular" classes contained approximately 24 students and "small" classes contained approximately 15 students.) Suppose that, in the population, the standardized tests have a mean score of 925 points and a standard deviation of 75 points. Let SmallClass denote a binary variable equal to 1 if the student is assigned to a small class and equal to zero otherwise. A regression of Testscore on SmallClass yeilds

TestScore = 918.0 + 13.9 × SmallClass, R2 = 0.01, SER = 74.6

(1.6) (2.5)

Standard errors are below the coefficients in the parentheses.

a. Do small classes improve test scores? By how much? Is the effect large? Explain.

b. Is the estimated effect of class size on test scores statistically significant? Carry out a test at the 5% significance level.

c. Construct a 99% confidence interval for the effect of SmallClass on Testscore.

d. Do you think that the regression errors are plausibly homoskedastic? Explain.

#### Solution Preview

a)

If a student is assigned to a large class (smallclass = 0), then his test score will be

TestScore = 918 + 13.9 X (0) = 918.

If assigned to a small class instead, TestScore = 918 + 13.9 X (1) = 931.9.

Small class does improve test score, by 13.9 points on average. The increment is roughly 13.9/918 = 1.5%, thus the effect is not very large.

b)

To perform a hypothesis test, we first write the null and alternative hypotheses. We use β for the coefficient of TestScore. βhat, which is the estimated value of β, is 13.9.

H0: β = 0 (i.e. no statistical difference between large and small classes).

Ha: β ≠ 0

Note that here we are testing TestScore ≠ 0, which means we are only interested in there is a difference (could be positive or negative difference). If the question says ...

#### Solution Summary

Hypothesis Tests & Confidence Intervals