Sketch a diagram that might be expected when $x$ and $y$ are related approximately by \(y=a

Question

Sketch a diagram that might be expected when $x$ and $y$ are related approximately by $y=a+b\ln x$ in each of the cases (i) and (ii) 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. $a$ and $b$ are both positive. [1] $a$ is positive and $b$ is negative.[1] In a chemical reaction, the yield of a chemical compound, $m$ grams, at different temperature ${{t}^{\circ }}$ C are given in the table below. $t$ $10$ $15$ $22$ $27$ $31$ $35$ $40$ $45$ $m$ $1.83$ $3.02$ $4.67$ $13.96$ $5.67$ $8.07$ $8.35$ $8.60$ Draw the scatter diagram for these values, labelling the axes clearly. [2] One of the pairs of data appears to be recorded wrongly. Indicate the corresponding point on your diagram by labelling it $P$, and explain why the scatter diagram for the remaining points may be consistent with a linear model. [2] For the rest of the questions, omit $P$ in your calculation. Calculate the equation of the least squares regression line of $m$ on $t$.[1] When the data $t=50$and $m=k$ was obtained and added to the seven pairs of data above, the new regression line is $m=0.16934t+0.72035$ correct to $5$ decimal places. Find the value of $k$, giving your answer in $2$ decimal places. [3]

Accepted Answer

(b)(ii) With the removal of the point $P$, as $t$ increases, $m$ increases at an approximately constant rate which is consistent with a linear model. With the removal of the point $P$, the remaining points lie close to a straight line which is consistent with a linear model. (b)(iii) $m=0.204t-0.0323$ (b)(iv) $k=7.55$

2015 MJC P2 Q9

2015 MJC P2 Q9