Data Relationships practice questions — SAT Math

Boys $=20-12=8$.

Fraction of boys $={\displaystyle \frac{8}{20}}={\displaystyle \frac{2}{5}}$.

The Further Maths row must add up to its own total, so $x=48-30=18$. The instrument column confirms this reading: $18+24=42$.

Choosing at random from the students who take Further Maths restricts attention to that row alone, so the denominator is the row total $48$, not the grand total $120$. Of those $48$ students, $18$ play an instrument.

$${\displaystyle \frac{18}{48}}={\displaystyle \frac{3}{8}}$$

Both given points lie on the line, so they fix its gradient: $$m=\frac{41-17}{16-4}=\frac{24}{12}=2.$$ Use the point $(4,17)$ to write the equation of the line: $$y-17=2(x-4),$$ $$y=2x-8+17=2x+9.$$ The temperature $13{}^{\circ }\mathrm{C}$ lies inside the range $3{}^{\circ }\mathrm{C}$ to $18{}^{\circ }\mathrm{C}$ covered by the data, so reading the line at $x=13$ is interpolation and the estimate is a legitimate one. Substituting $x=13$: $$y=2(13)+9=26+9=35.$$

Row Y total $=20$. Grand total $=50$.

Fraction $={\displaystyle \frac{20}{50}}={\displaystyle \frac{2}{5}}$.

Read off the pattern in the readings that agree with one another: $y$ rises by $3$ for each step of $1$ in $x$, and at $x=1$ the height is $4$, so the trend line has gradient $3$ and passes through $(1,4)$: $$y=3x+1.$$ Compare the height predicted by this line with the height actually recorded: $$\begin{array}{cccccccc}x& 1& 2& 3& 4& 5& 6& 7\\ 3x+1& 4& 7& 10& 13& 16& 19& 22\\ y& 4& 7& 10& 5& 16& 19& 22\end{array}$$ Six of the seven readings sit exactly on the line, so their vertical gaps are $0$. At $x=4$ the line predicts $3(4)+1=13$ while the recorded height is $5$, a vertical gap of $$13-5=8.$$ That gap is far larger than any other, so the reading that breaks the trend is $$(4,5)\mathrm{.}$$

An upward (rising-left-to-right) trend means higher $x$ is associated with higher $y$ — a positive correlation.

The slope is $8$ score points per additional hour. For a $0.5$-hour increase, the predicted change is $0.5\times 8=4$.