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| 1 | $(\mathrm{IB} / \mathrm{sl} / 2019 /$ November $/$ Paper $2 / \mathrm{q} 1)$
[Maximum mark: 6] The number of messages, $M$, that six randomly selected teenagers sent during the month of October is shown in the following table. The table also shows the time, $T$, that they spent talking on their phone during the same month. $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text{Time spent talking}&&&&&&\\ \text{on their phone (T minutes)} & 50 & 55 & 105 & 128 & 155 & 200 \\\hline \text{Number of messages (M)} & 358 & 340 & 740 & 731 & 800 & 992 \\\hline\end{array}$$ The relationship between the variables can be modelled by the regression equation $M=a T+b$.
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| 2 | (IB/sl/2019/May/paper2tz1/q5)
[Maximum mark: 6] A jigsaw puzzle consists of many differently shaped pieces that fit together to form a picture.  $$\begin{array}{|l|r|r|r|r|r|r|}\hline \text{Edge pieces $(x)$} & 16 & 31 & 39 & 55 & 84 & 115 \\\hline \text{Interior pieces $(y)$} & 89 & 239 & 297 & 402 & 580 & 802 \\\hline\end{array}$$ Jill models the relationship between these variables using the regression equation $y=a x+b$.
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| 3 | (IB/sl/2019/May/paper2tz2/q1)
[Maximum mark: 6] A group of 7 adult men wanted to see if there was a relationship between their Body Mass Index (BMI) and their waist size. Their waist sizes, in centimetres, were recorded and their BMI calculated. The following table shows the results. $$\begin{array}{|l|c|c|c|c|c|c|c|}\hline \text{Waist $(x \mathrm{~cm})$} & 58 & 63 & 75 & 82 & 93 & 98 & 105 \\\hline \text{BMI $(y)$ }& 19 & 20 & 22 & 23 & 25 & 24 & 26 \\\hline\end{array}$$ The relationship between $x$ and $y$ can be modelled by the regression equation $y=a x+b$.
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| 4 | (IB/sl/2018/November/Paper2/q2)
[Maximum mark: 6] The following table shows the hand lengths and the heights of five athletes on a sports team. $$\begin{array}{|c|r|r|r|r|r|}\hline \text{Hand length $(x \mathrm{~cm})$} & 21.0 & 21.9 & 21.0 & 20.3 & 20.8 \\\hline \text{Height $(y \mathrm{~cm})$ }& 178.3 & 185.0$ & 177.1 & 169.0 & 174.6 \\\hline\end{array}$$ The relationship between $x$ and $y$ can be modelled by the regression line with equation $y=a x+b$.
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| 5 | (IB/sl/2018/May/paper2tz1/q8)
[Maximum mark: 13] The following table shows values of $\ln x$ and $\ln y$. $$\begin{array}{|l|l|l|l|l|}\hline \ln x & 1.10 & 2.08 & 4.30 & 6.03 \\\hline \ln y & 5.63 & 5.22 & 4.18 & 3.41 \\\hline\end{array}$$ The relationship between $\ln x$ and $\ln y$ can be modelled by the regression equation $\ln y=a \ln x+b$
The relationship between $x$ and $y$ can be modelled using the formula $y=k x^{\prime \prime}$. where $k \neq 0, n \neq 0, n \neq 1$.
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| 6 | (IB/sl/2018/May/paper2tz2/q1)
[Maximum mark: 6] The following table shows the mean weight, $y \mathrm{~kg}$, of children who are $x$ years old. $$\begin{array}{|c|c|c|c|c|c|}\hline \text{Age ( $x$ years) }& 1.25 & 2.25 & 3.5 & 4.4 & 5.85 \\\hline \text{Weight $(y \mathrm{~kg})$} & 10 & 13 & 14 & 17 & 19 \\\hline\end{array}$$ The relationship between the variables is modelled by the regression line with equation $y=a x+b$.
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| 7 | (IB/sl/2017/November/Paper2/g8)
[Maximum mark: 14] Adam is a beekecper who collected data about monthly honcy production in his bee hives. The data for six of his hives is shown in the following table. $$\begin{array}{|c|c|c|c|c|c|c|}\hline \text{Number of bees $(N)$} & 190 & 220 & 250 & 285 & 305 & 320 \\\hline \text{Monthly honey production in grams $(P)$} & 900 & 1100 & 1200 & 1500 & 1700 & 1800 \\\hline\end{array}$$ The relationship between the variables is modelled by the regression line with equation $P=a N+b$.
Adam has 200 hives in total. He collects data on the monthly honey production of all the hives. This data is shown in the following cumulative frequency graph.  |
| 8 | Question (IB/sl/2017/November/Paper2/q8b) 7 continued
Adam's hives are labelled as low, regular or high production, as defined in the following table. $$\begin{array}{|l|c|c|l|}\hline \text{Type of hive }&\text{ low} & \text{regular} & \text{high} \\\hline \text{Monthly honey production}&&&\\ \text{ in grams $(P)$} & P \leq 1080 & 1080 < P \leq k & P>k \\\hline\end{array}$$
Adam knows that 128 of his hives have a regular production.
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| 9 | (IB/sl/2017/May/paper1tz1/q4)
[Maximum mark: 6] Jim heated a liquid until it boiled. He measured the temperature of the liquid as it cooled. The following table shows its temperature, $d$ degrees Celsius, $I$ minutes after it boiled. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t(\min ) & 0 & 4 & 8 & 12 & 16 & 20 \\\hline d\left({ }^{\circ} \mathrm{C}\right) & 105 & 98.4 & 85.4 & 74.8 & 68.7 & 62.1 \\\hline\end{array}$$
Jim believes that the relationship between $d$ and $t$ can be modelled by a linear regression equation.
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| 10 | (IB/sl/2017/May/paper2tz2/q2)
[Maximum mark: 7] The maximum temperature $T$, in degrees Celsius, in a park on six randomly selected days is shown in the following table. The table also shows the number of visitors, $N$, to the park on each of those six days. $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text{Maximum temperature $(T)$} & 4 & 5 & 17 & 31 & 29 & 11 \\\hline \text{Number of visitors $(N)$} & 24 & 26 & 36 & 38 & 46 & 28 \\\hline\end{array}$$ The relationship between the variables can be modelled by the regression equation $N=a T+b$.
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| 11 | (IB/sl/2016/May/paper2tz1/q5)
[Maximum mark: 6] The mass $M$ of a decaying substance is measured at one minute intervals. The points $(t, \ln M)$ are plotted for $0 \leq t \leq 10$, where $t$ is in minutes. The line of best fit is drawn. This is shown in the following diagram.  The correlation coefficient for this linear model is $r=-0.998$.
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| 12 | (IB/sl/2016/May/paper2tz2/q8)
[Maximum mark: 15] The price of a used car depends partly on the distance it has travelled. The following table shows the distance and the price for seven cars on 1 January $2010 .$ $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline \text{Distance, $x \mathrm{~km}$ }& 11500 & 7500 & 13600 & 10800 & 9500 & 12200 & 10400 \\\hline \text{Price, y dollars} & 15000 & 21500 & 12000 & 16000 & 19000 & 14500 & 17000 \\\hline\end{array}$$ The relationship between $x$ and $y$ can be modelled by the regression equation $y=a x+b$.
On 1 January $2010 .$ Lina buys a car which has travelled $1100 \mathrm{~km}$.
The price of a car decreases by $5 \%$ each year.
Lina will sell her car when its price reaches 10000 dollars.
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| 13 | (IB/sl/2015/November/Paper2/q9)
[Maximum mark: 16] An environmental group records the numbers of coyotes and foxes in a wildlife reserve after $t$ years, starting on 1 January 1995 . Let $c$ be the number of coyotes in the reserve after $t$ years. The following table shows the number of coyotes after $t$ years. $$\begin{array}{|c|c|c|c|c|c|}\hline \text{number of years} (t) & 0 & 2 & 10 & 15 & 19 \\\hline \text{number of coyotes} (c)& 115 & 197 & 265 & 320 & 406 \\\hline\end{array}$$ The relationship between the variables can be modelled by the regression equation $c=a t+b$
Let $f$ be the number of foxes in the reserve after $t$ years. The number of foxes can be modelled by the equation $f=\frac{2000}{1+99 \mathrm{e}^{-k t}}$, where $k$ is a constant.
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| 14 | (IB/sl/2015/May/paper2tz1/q1)
[Maximum mark: 7] The following table shows the average number of hours per day spent watching television by seven mothers and each mother's youngest child. $$\begin{array}{|l|c|c|c|c|c|c|c|}\hline\text{Hours per day that}&&&&&&&\\ \text{ a mother watches television}(x) & 2.5 & 3.0 & 3.2 & 3.3 & 4.0 & 4.5 & 5.8 \\\hline \text{Hours per day that}&&&&&&&\\ \text{ her child watches television}(y) & 1.8 & 2.2 & 2.6 & 2.5 & 3.0 & 3.2 & 3.5 \\\hline\end{array}$$ The relationship can be modelled by the regression line with equation $y=a x+b$.
Elizabeth watches television for an average of $3.7$ hours per day.
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| 15 | (IB/sl/2015/May/paper2tz2/q3)
[Maximum mark: 6] The following table shows the sales, $y$ millions of dollars, of a company, $x$ years after it opened. $$\begin{array}{|l|c|c|c|c|c|}\hline \text{Time after opening ( $x$ years)} & 2 & 4 & 6 & 8 & 10 \\\hline \text{Sales ( $y$ millions of dollars)} & 12 & 20 & 30 & 36 & 52 \\\hline\end{array}$$ The relationship between the variables is modelled by the regression line with equation $y=a x+b$.
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Answer ( Regression)
[1](a) $a=4.30,b-163$ (b) 826
[2](a) $a=6.93,b=8.81$ (b) 93
[3](a)(i) $a=0.141, b=11.1$ (ii) $r=0.978$ (b) $24.5$
[4](a)(i) $a=9.91, b=-31.3$ (ii) $r=0.986$ (b) 182
[5](a) $a=-0.454, b=6.14$ (b) $y=261$ (c) $n=-0.454, k=465$
[6](a)(i) $a=1,92, b=7.98$ (ii) $r=0.985$ (b) $11.7$
[7](a) $a=6.96, b=-455$ (b) $p=1420$
[8](c) 40 (d) (i) $k=1640$ (ii) 32 (e) $0.144$
[9](a) (i) $t$ (ii) 105 (b) $-0.992$ (c) $4.48$
[10](a)(i) $\quad a=0.667, b=22.2$ (ii) $r=0.923$ (b) 32
[11](a) Strong, Negative (b) $b=e^{-0.12}$
[12](a)(i) $r=-0.994$ (ii) $a=-1.58, b=33500$ (b) 16100 (c) 11800 (d) 2019
[13](a) $\quad a=13.4, b=137$ (b) 231 (c) 20 (d) $k=-\frac{1}{5} \ln \left(\frac{11}{36}\right)$ (e) 2007
[14] (a) (i) $r=0.947$ (ii) $a=0.501, b=0.804$ (b) $2.7$
[15] (a) (i) $a=4.8, b=1.2$ (ii) $r=0.988$ (b) $34.8$
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