RELATIONSHIP BETWEEN THE STUDENT-TEACHER RATIO AND

AVERAGE TEST SCORES

James Skinner March 25, 2016

The California School District has

provided 420 pieces of data for 12 different

variables. These variables include total

enrollment, number of teachers, number of

computers, and many other important pieces of

information. Of these 420 data sets, we will

select a random sub-sample of half this amount,

or n = 210. Next we will compute the summary

statistics of the variables average test score

(testscr) and student teacher ratio (str). Refer to

Table 1 below to view the results.

Summary Statistics, using the observations 1 - 210

MEAN MEDIAN MINIMUM

str 656.401 656.6 615.75

testscr 19.610 19.802 14.202

MAXIMUM STD. DEV

EX KURTOSIS

str 706.75 18.223 -0.2356

testscr 24.889 1.881 0.4156 Table 1: Summary Statitstics of 'Average Test Scores'

and 'Student Teacher Ratio'

After testing the correlation between

these two variables, we discover:

corr(testscr, str) = -0.23148045

And through a two-tailed T-test:

Under the null hypothesis of no correlation:

t(208) = -3.43166, with two-tailed p-value

0.0007

The P-value is evidence that correlation

between average test scores and the student

teacher ratio is significantly different than zero.

The Regression Model

The regression model for this data will be as

follows:

testscr = α + β(stri) + µi ; i = 1,…210

And we will assume that

1) µi has conditional mean of zero given Xi

: E(µi|Xi) = 0

2) (Xi, Yi), i = 1,…,n are identical and are

identically distributed iid draws from

their distribution

3) Large outliers are unlikely: Xi and Yi

have nonzero finite fourth moments

RESULTS OF OLS MODEL ESTIMATE

The results of our estimation of the

model are as follows:

What these results indicate is that the

student to teacher ratio (str) has a negative

effect on average test scores (testscr).

Specifically, if you were to increase the

student to teacher ratio by one unit, then

you would expect to see

100 * (-2.242 / 656.401) = -0.342%

which represents a -0.342% change in

average test scores.

We have taken the data and plotted in

on a scatter-plot, which can be seen in

figure 1 below. This graph indicates that

there is a negative relationship between

student teacher ratios and average test

scores. The arrow points to the plot (19.61,

656.40), which is the sample mean of both

variables. This information can be seen

above in Table 1.

Figure 1: Scatter-plot of Average Test Scores (Y axis) vs. Student-Teacher Ratio (X axis)

To confirm that our estimation results were

correct, we will test:

18.223 / 1.881 * (-2.231) = -2.242

This proves that the estimated amounts, since

the variance is identical and independent.

Summary

The goal of this report was to explain

the results of several tests on average test

scores and the student-teacher ratio. We

wanted to quantify the relationship and

interpret the regression analysis of the data.

By computing the summary statistics

and the correlation between the two variables,

we learned that there is a negative relationship

between the two. Delving farther in, the

regression model was created to test the

results. With this, we gathered that a one unit

increase in the student-teacher ratio will result

in a -0.342% change in the average test scores.

By plotting the data on a scatter plot,

we were able to visualize the negative

relationship we predicted. The data is

correlated. We confirmed our estimates by

analyzing the standard deviation and the

correlation. This lined up and proved that our

estimates were correct.