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4.2 Estimating the Coefficients of the Linear Regression Model 161

TABLE 4.1 Summary of the Distribution of Student–Teacher Ratios and Fifth-Grade
Test Scores for 420 K–8 Districts in California in 1999

Percentile

Average Standard 10% 25% 40% 50% 60% 75% 90%
Student–teacher ratio 19.6 Deviation 19.3 (median) 20.1
17.3 18.6 20.9 21.9
1.9 630.4 640.0 19.7 666.7 679.1

Test score 654.2 19.1 649.1 654.5 659.4

student–teacher ratio is 17.3 (that is, only 10% of districts have student–teacher
ratios below 17.3), while the district at the 90th percentile has a student–teacher
ratio of 21.9.

A scatterplot of these 420 observations on test scores and the student–teacher
ratio is shown in Figure 4.2. The sample correlation is -0.23, indicating a weak
negative relationship between the two variables. Although larger classes in this
sample tend to have lower test scores, there are other determinants of test scores
that keep the observations from falling perfectly along a straight line.

Despite this low correlation, if one could somehow draw a straight line
through these data, then the slope of this line would be an estimate of bClassSize

Figure 4.2 Scatterplot of Test Score vs. Student–Teacher Ratio (California School District Data)

Data from 420 Test score
California school dis- 720
tricts. There is a weak 700
negative relationship 680
between the student– 660
teacher ratio and test
scores: The sample
correlation is - 0.23.

640

620

60010 15 20 25 30
Student–teacher ratio