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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. $$\begin{array}{lcccccc}\hline \text{Time spent talking}&&&&&&\\ \text{on their phone (T minutes)} & 50 & 55 & 105 & 128 & 155 & 200 \\\hline \text{Number of messages (M)} & 358 & 340 & 740 & 731 & 800 & 992 \\\hline\end{array}$$ The relationship between the variables can be modelled by the regression equation $M=a T+b$.

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. Jill is doing a 1000 piece jiggaw puzzle. She started by sorting the edge pieces from the interior pieces. Six times she stopped and counted how many of each type she had found. The following table indicates this information. $$\begin{array}{lrrrrrr}\hline \text{Edge pieces $(x)$} & 16 & 31 & 39 & 55 & 84 & 115 \\\hline \text{Interior pieces $(y)$} & 89 & 239 & 297 & 402 & 580 & 802 \\\hline\end{array}$$ Jill models the relationship between these variables using the regression equation $y=a x+b$.

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. $$\begin{array}{lccccccc}\hline \text{Waist $(x \mathrm{~cm})$} & 58 & 63 & 75 & 82 & 93 & 98 & 105 \\\hline \text{BMI $(y)$ }& 19 & 20 & 22 & 23 & 25 & 24 & 26 \\\hline\end{array}$$ The relationship between $x$ and $y$ can be modelled by the regression equation $y=a x+b$.

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. $$\begin{array}{crrrrr}\hline \text{Hand length $(x \mathrm{~cm})$} & 21.0 & 21.9 & 21.0 & 20.3 & 20.8 \\\hline \text{Height $(y \mathrm{~cm})$ }& 178.3 & 185.0$ & 177.1 & 169.0 & 174.6 \\\hline\end{array}$$ The relationship between $x$ and $y$ can be modelled by the regression line with equation $y=a x+b$.

5  (IB/sl/2018/May/paper2tz1/q8)
[Maximum mark: 13] The following table shows values of $\ln x$ and $\ln y$. $$\begin{array}{lllll}\hline \ln x & 1.10 & 2.08 & 4.30 & 6.03 \\\hline \ln y & 5.63 & 5.22 & 4.18 & 3.41 \\\hline\end{array}$$ 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 $k \neq 0, n \neq 0, n \neq 1$.

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. $$\begin{array}{cccccc}\hline \text{Age ( $x$ years) }& 1.25 & 2.25 & 3.5 & 4.4 & 5.85 \\\hline \text{Weight $(y \mathrm{~kg})$} & 10 & 13 & 14 & 17 & 19 \\\hline\end{array}$$ The relationship between the variables is modelled by the regression line with equation $y=a x+b$.

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. $$\begin{array}{ccccccc}\hline \text{Number of bees $(N)$} & 190 & 220 & 250 & 285 & 305 & 320 \\\hline \text{Monthly honey production in grams $(P)$} & 900 & 1100 & 1200 & 1500 & 1700 & 1800 \\\hline\end{array}$$ 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. $$\begin{array}{lccl}\hline \text{Type of hive }&\text{ low} & \text{regular} & \text{high} \\\hline \text{Monthly honey production}&&&\\ \text{ in grams $(P)$} & P \leq 1080 & 1080 < P \leq k & P>k \\\hline\end{array}$$
Adam knows that 128 of his hives have a regular production.

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. $$\begin{array}{ccccccc}\hline t(\min ) & 0 & 4 & 8 & 12 & 16 & 20 \\\hline d\left({ }^{\circ} \mathrm{C}\right) & 105 & 98.4 & 85.4 & 74.8 & 68.7 & 62.1 \\\hline\end{array}$$
Jim believes that the relationship between $d$ and $t$ can be modelled by a linear regression equation.

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. $$\begin{array}{lcccccc}\hline \text{Maximum temperature $(T)$} & 4 & 5 & 17 & 31 & 29 & 11 \\\hline \text{Number of visitors $(N)$} & 24 & 26 & 36 & 38 & 46 & 28 \\\hline\end{array}$$ The relationship between the variables can be modelled by the regression equation $N=a T+b$.

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 $r=0.998$.

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 $2010 .$ $$\begin{array}{llllllll}\hline \text{Distance, $x \mathrm{~km}$ }& 11500 & 7500 & 13600 & 10800 & 9500 & 12200 & 10400 \\\hline \text{Price, y dollars} & 15000 & 21500 & 12000 & 16000 & 19000 & 14500 & 17000 \\\hline\end{array}$$ 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.

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. $$\begin{array}{cccccc}\hline \text{number of years} (t) & 0 & 2 & 10 & 15 & 19 \\\hline \text{number of coyotes} (c)& 115 & 197 & 265 & 320 & 406 \\\hline\end{array}$$ 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.

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. $$\begin{array}{lccccccc}\hline\text{Hours per day that}&&&&&&&\\ \text{ a mother watches television}(x) & 2.5 & 3.0 & 3.2 & 3.3 & 4.0 & 4.5 & 5.8 \\\hline \text{Hours per day that}&&&&&&&\\ \text{ her child watches television}(y) & 1.8 & 2.2 & 2.6 & 2.5 & 3.0 & 3.2 & 3.5 \\\hline\end{array}$$ 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.

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. $$\begin{array}{lccccc}\hline \text{Time after opening ( $x$ years)} & 2 & 4 & 6 & 8 & 10 \\\hline \text{Sales ( $y$ millions of dollars)} & 12 & 20 & 30 & 36 & 52 \\\hline\end{array}$$ The relationship between the variables is modelled by the regression line with equation $y=a x+b$.

Answer ( Regression)
[1](a) $a=4.30,b163$ (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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