Sketch a scatter diagram that might be expected when $x$ and $y$ are related approximately as gi

Question

Sketch a scatter diagram that might be expected when $x$ and $y$ are related approximately as given in each of the cases (A) and (B) below. In each case your diagram should include $6$ points, approximately equally spaced with respect to $x$, and with all $x$- and $y$- values positive. The letters $a$, $b$, $c$ and $d$ represent constants.[2] $y = a + bx^2$, where $a$ is positive and $b$ is negative, $y = c + \frac{\mathrm{d}}{x}$, where $c$ is positive and $d$ is positive. Alvin is investigating the relationship between the amount of screen time per day and the sleep quality of youths. He recorded the sleep quality score, $y$, on a scale of $1$ to $10$ with $1$ being the worst and $10$ being the best, of particular youths with $x$ hours of screen time per day. x 1.6 1.7 1.9 2.0 2.5 2.8 3.9 y 9 8 7 6 4 3 1 Draw a scatter diagram for these values, labelling the axes.[1] Explain which of the two cases in part $ (i) $ is the most appropriate for modelling these values and calculate the product moment correlation coefficient for this case.[2] Alvin wants to estimate the sleep quality score when the screen time per day is $ 3 $ hours. Use the case that you have identified in part $ (iii) $ to find the equation of a suitable regression line and use your equation to find the required estimate. Explain whether you would expect this estimate to be reliable.[4] Write the equation from part $ (iv) $ in terms of $ y $ and $ t $, where $ t $ is the amount of screen time per day in minutes. Explain whether the product moment correlation coefficient for this case differs from the one found in part $ (iii) $.[3] Alvin concluded that this study shows that increased amount of screen time per day causes poorer sleep quality. Explain whether you agree with his conclusion.[1]

Accepted Answer

(iii)r=0.999 (iv)$y=-4.66+\frac{21.7}{x}$, y= 3 when x =3 (v)$y=-4.66+\frac{1300}{t}$

2025 SAJC P2 Q9

2025 SAJC P2 Q9