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.
1. $a$ and $b$ are both positive. [1]
2. $a$ 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.
$t$
$10$
$15$
$22$
$27$
$31$
$35$
$40$
$45$
$m$
$1.83$
$3.02$
$4.67$
$13.96$
$5.67$
$8.07$
$8.35$
$8.60$
1. Draw the scatter diagram for these values, labelling the axes clearly. [2]
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.
3. Calculate the equation of the least squares regression line of $m$ on $t$.[1]
4. 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

(a)(i) Increasing, flattening six-point pattern. (ii) Decreasing, flattening six-point pattern; all coordinates positive. (b)(i) Scatter shown. (ii) $P=(27,13.96)$. (iii) $m\approx0.204t-0.0323$. (iv) $k\approx7.55$ g.

2015 MJC P2 Q9

2015 MJC P2 Q9

Solution:

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