Improving SAT scores.The National Education Longitudinal Survey (NELS) tracks a nationally representative sample of U.S. students from eighth grade through high school and college. Research published in Chance (Winter 2001) examined the Standardized Assessment Test (SAT) scores of 265 NELS students who paid a private tutor to help them improve their scores. The table summarizes the changes in both the SAT–Mathematics and SAT–Verbal scores for these students.

	SAT–Math	SAT–Verbal
Mean change in score	19	7
Standard deviation of score changes	65	49

a.Suppose one of the 265 students who paid a private tutor is selected at random. Give an interval that is likely to contain this student’s change in the SAT–Math score.

b.Repeat part afor the SAT–Verbal score.

c.Suppose the selected student increased his score on one of the SAT tests by 140 points. Which test, the SAT– Math or SAT–Verbal, is the one most likely to have the 140-point increase? Explain.

We do not know the distribution of the scores; therefore, we will use the Chebyshev rule to find the interval.

As per the rule, 8/9 proportion, i.e., almost 89% of all the students’ scores, will fall under the 3rd standard deviation from the mean. Hence, we use $\overline{x}\pm 3s$to find the interval.

$$\begin{gathered} \overline{x}\pm 3s \\ Lowerrange=\overline{x}-3s \\ =19-3\left(65\right) \\ =19-195 \\ =-176 \\ Upperrange=\overline{x}+3s \\ =19+3\left(65\right) \\ =19+195 \\ =214 \end{gathered}$$

Therefore, the most likely chances are that a random student’s SAT-Math score will be anywhere from 176 below his/her previous SAT-Math score to 214 above his/her SAT-Math score.

We will be using the Chebyshev again rule due to similar reasoning,

$$\begin{gathered} \overline{x}\pm 3s \\ Lowerrange=\overline{x}-3s \\ =7-3\left(49\right) \\ =7-147 \\ =-140 \\ Upperrange=\overline{x}+3s \\ =7+3\left(49\right) \\ =7+147 \\ =154 \end{gathered}$$

Therefore, the interval where a random student’s SAT-Verbal score might fall is 140 below his/her previous score to 154 above his/her previous score.

An increase of 140 points for SAT-Math would be a little less than 2 standard deviations from the mean, whereas a 140 point increase in SAT-Verbal would be a little less than 3 standard deviations from the mean. As a 140 point change in SAT-Math is not as large as a 140 point change in SAT-Verbal, it is most likely an SAT-Math score.