Sukhila is a biologist investigating the relationship between the amount of growth hormone, x grams, supplied to an orchid plant, and the number of flowers ( y ) it produces. x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 Sukhila models the data using the y on x regression line with equation y = a x + b Write down the values of a and of b ,

Given data: x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 From the given table we can find the following values, Serial number The amount of growth hormone x The number of flowers produce y x 2 x y 1 1 6 1 6 2 1 . 5 7 2 . 25 10 . 5 3 2 10 4 20 4 2 . 5 15 6 . 25 37 . 5 5 3 9 9 27 6 3 . 5 17 12 . 25 59 . 5 7 4 20 16 80 8 4 . 5 18 20 . 25 81 Total ∑ x = 22 ∑ y = 102 ∑ x 2 = 71 ∑ x y = 321 . 5 The line of regression of y on x y = b + a x ∴ ∑ y = n b + a ∑ x ⇒ 102 = 8 b + 22 a ------(1) ∴ ∑ x y = b ∑ x + a ∑ x 2 ⇒ 321 . 5 = 22 b + 71 a ------(2) Solve equation (1) and (2), a = 3 . 905 and b = 2 . 012 Thus, the line of regression y = 3 . 905 x + 2 . 012 .

Sukhila is a biologist investigating the relationship between the amount of growth hormone, x grams, supplied to an orchid plant, and the number of flowers ( y ) it produces. x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 Sukhila models the data using the y on x regression line with equation y = a x + b Explain what the gradient represents.

Given data: x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 From the given table we can find the following values, Serial number The amount of growth hormone x The number of flowers produce y x 2 x y 1 1 6 1 6 2 1 . 5 7 2 . 25 10 . 5 3 2 10 4 20 4 2 . 5 15 6 . 25 37 . 5 5 3 9 9 27 6 3 . 5 17 12 . 25 59 . 5 7 4 20 16 80 8 4 . 5 18 20 . 25 81 Total ∑ x = 22 ∑ y = 102 ∑ x 2 = 71 ∑ x y = 321 . 5 The line of regression of y on x y = b + a x ∴ ∑ y = n b + a ∑ x ⇒ 102 = 8 b + 22 a ------(1) ∴ ∑ x y = b ∑ x + a ∑ x 2 ⇒ 321 . 5 = 22 b + 71 a ------(2) Solve equation (1) and (2), a = 3 . 905 and b = 2 . 012 Thus, the line of regression y = 3 . 905 x + 2 . 012 . Here, the value of gradient is a = 3 . 905 We know that, the equation of regression line is y = a + b x , where b is the slope of the line and a is the y -intercept. Slope of regression line or gradient represents the change in y as x changes, if y is dependent of x . The intercept of the regression line is just the predicted value for y , when x is 0 . Thus, an increase of 2 g of growth hormone produces nearly 4 extra flowers.

Sukhila is a biologist investigating the relationship between the amount of growth hormone, x grams, supplied to an orchid plant, and the number of flowers ( y ) it produces. x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 Sukhila models the data using the y on x regression line with equation y = a x + b Explain what the y -intercept represents.

Given data: x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 From the given table we can find the following values, Serial number The amount of growth hormone x The number of flowers produce y x 2 x y 1 1 6 1 6 2 1 . 5 7 2 . 25 10 . 5 3 2 10 4 20 4 2 . 5 15 6 . 25 37 . 5 5 3 9 9 27 6 3 . 5 17 12 . 25 59 . 5 7 4 20 16 80 8 4 . 5 18 20 . 25 81 Total ∑ x = 22 ∑ y = 102 ∑ x 2 = 71 ∑ x y = 321 . 5 The line of regression of y on x y = b + a x ∴ ∑ y = n b + a ∑ x ⇒ 102 = 8 b + 22 a ------(1) ∴ ∑ x y = b ∑ x + a ∑ x 2 ⇒ 321 . 5 = 22 b + 71 a ------(2) Solve equation (1) and (2), a = 3 . 905 and b = 2 . 012 Thus, the line of regression y = 3 . 905 x + 2 . 012 . Here, the value of gradient is b = 2 . 012 We know that, the equation of regression line is y = a + b x , where b is the slope of the line and a is the y -intercept. Slope of regression line or gradient represents the change in y as x changes, if y is dependent of x . The intercept of the regression line is just the predicted value for y , when x is 0 . Thus, a plant with no growth hormone would have 2 flowers.

Sukhila is a biologist investigating the relationship between the amount of growth hormone, x grams, supplied to an orchid plant, and the number of flowers ( y ) it produces. x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 Sukhila models the data using the y on x regression line with equation y = a x + b Estimate the number of flowers that Sukhila would expect from 1 . 75 grams of growth hormone.

Given data: x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 From the given table we can find the following values, Serial number The amount of growth hormone x The number of flowers produce y x 2 x y 1 1 6 1 6 2 1 . 5 7 2 . 25 10 . 5 3 2 10 4 20 4 2 . 5 15 6 . 25 37 . 5 5 3 9 9 27 6 3 . 5 17 12 . 25 59 . 5 7 4 20 16 80 8 4 . 5 18 20 . 25 81 Total ∑ x = 22 ∑ y = 102 ∑ x 2 = 71 ∑ x y = 321 . 5 The line of regression of y on x y = b + a x ∴ ∑ y = n b + a ∑ x ⇒ 102 = 8 b + 22 a ------(1) ∴ ∑ x y = b ∑ x + a ∑ x 2 ⇒ 321 . 5 = 22 b + 71 a ------(2) Solve equation (1) and (2), a = 3 . 905 and b = 2 . 012 Thus, the line of regression y = 3 . 905 x + 2 . 012 . Put, x = 1 . 75 , y = 3 . 905 × 1 . 75 + 2 . 012 Thus, number of flowers are 8 . 85 flowers.

Sukhila is a biologist investigating the relationship between the amount of growth hormone, x grams, supplied to an orchid plant, and the number of flowers ( y ) it produces. x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 Sukhila models the data using the y on x regression line with equation y = a x + b Sukhila wanted the plant to produce 12 flowers. Find how much growth hormone she should use.

Given data: x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 From the given table we can find the following values, Serial number The amount of growth hormone x The number of flowers produce y x 2 x y 1 1 6 1 6 2 1 . 5 7 2 . 25 10 . 5 3 2 10 4 20 4 2 . 5 15 6 . 25 37 . 5 5 3 9 9 27 6 3 . 5 17 12 . 25 59 . 5 7 4 20 16 80 8 4 . 5 18 20 . 25 81 Total ∑ x = 22 ∑ y = 102 ∑ x 2 = 71 ∑ x y = 321 . 5 The line of regression of y on x y = b + a x ∴ ∑ y = n b + a ∑ x ⇒ 102 = 8 b + 22 a ------(1) ∴ ∑ x y = b ∑ x + a ∑ x 2 ⇒ 321 . 5 = 22 b + 71 a ------(2) Solve equation (1) and (2), a = 3 . 905 and b = 2 . 012 Thus, the line of regression y = 3 . 905 x + 2 . 012 . Put y = 12 , ⇒ 12 = 3 . 905 × x + 2 . 012 ⇒ x = 2 . 55 grams. Thus, hormone she should use is 2 . 55 grams.

Sukhila is a biologist investigating the relationship between the amount of growth hormone, x grams, supplied to an orchid plant, and the number of flowers ( y ) it produces. x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 Sukhila models the data using the y on x regression line with equation y = a x + b Explain why it might be inappropriate to use this model to predict the number of flowers when the plant is treated with 1 kg of growth hormone.

Given data: x 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 y 6 7 10 15 9 17 20 18 From the given table we can find the following values, Serial number The amount of growth hormone x The number of flowers produce y x 2 x y 1 1 6 1 6 2 1 . 5 7 2 . 25 10 . 5 3 2 10 4 20 4 2 . 5 15 6 . 25 37 . 5 5 3 9 9 27 6 3 . 5 17 12 . 25 59 . 5 7 4 20 16 80 8 4 . 5 18 20 . 25 81 Total ∑ x = 22 ∑ y = 102 ∑ x 2 = 71 ∑ x y = 321 . 5 The line of regression of y on x y = b + a x ∴ ∑ y = n b + a ∑ x ⇒ 102 = 8 b + 22 a ------(1) ∴ ∑ x y = b ∑ x + a ∑ x 2 ⇒ 321 . 5 = 22 b + 71 a ------(2) Solve equation (1) and (2), a = 3 . 905 and b = 2 . 012 Thus, the line of regression y = 3 . 905 x + 2 . 012 . Here, 1 kg is outside of the domain and extrapolation is unreliable. Thus, it is unreliable.

EASY

Diploma

IMPORTANT

Earn 100

The amount of fuel $(y$ litres $)$ used by a car to travel $x km$ is shown in this table.

Distance $(x km)$ Amount of fuel $(in y$ litres $)$

$30$ $3.2$

$65$ $7.5$

$110$ $8.4$

$140$ $15.1$

$185$ $16.5$

This data can be modelled by the regression line with equation $y = a x + b$
Explain what the gradient, $a$ , represents.

Important Questions on Modelling Relationships between Two Data Sets: Statistics for Bivariate Data

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 7

The amount of fuel $(y$ litres $)$ used by a car to travel $x km$ is shown in this table.

Distance $(x km)$	Amount of fuel $(in y$ litres $)$
$30$	$3.2$
$65$	$7.5$
$110$	$8.4$
$140$	$15.1$
$185$	$16.5$

This data can be modelled by the regression line with equation $y = a x + b$

Use the model to estimate the amount of fuel the car will use when it is driven for $160 km$ .

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 7

The amount of fuel $(y$ litres $)$ used by a car to travel $x km$ is shown in this table.

Distance $(x km)$	Amount of fuel $(in y$ litres $)$
$30$	$3.2$
$65$	$7.5$
$110$	$8.4$
$140$	$15.1$
$185$	$16.5$

This data can be modelled by the regression line with equation $y = a x + b$

Explain why it would be unreliable to use this model to predict the amount of fuel used on a journey of

5 km

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 8

Sukhila is a biologist investigating the relationship between the amount of growth hormone, $x$ grams, supplied to an orchid plant, and the number of flowers $(y)$ it produces.

$x$	$1$	$1.5$	$2$	$2.5$	$3$	$3.5$	$4$	$4.5$
$y$	$6$	$7$	$10$	$15$	$9$	$17$	$20$	$18$

Sukhila models the data using the $y$ on $x$ regression line with equation $y = a x + b$

Write down the values of $a$ and of $b$ ,

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 8

Sukhila is a biologist investigating the relationship between the amount of growth hormone, $x$ grams, supplied to an orchid plant, and the number of flowers $(y)$ it produces.

$x$	$1$	$1.5$	$2$	$2.5$	$3$	$3.5$	$4$	$4.5$
$y$	$6$	$7$	$10$	$15$	$9$	$17$	$20$	$18$

Sukhila models the data using the $y$ on $x$ regression line with equation $y = a x + b$

Explain what the gradient represents.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 8

Sukhila is a biologist investigating the relationship between the amount of growth hormone, $x$ grams, supplied to an orchid plant, and the number of flowers $(y)$ it produces.

$x$	$1$	$1.5$	$2$	$2.5$	$3$	$3.5$	$4$	$4.5$
$y$	$6$	$7$	$10$	$15$	$9$	$17$	$20$	$18$

Sukhila models the data using the $y$ on $x$ regression line with equation $y = a x + b$

Explain what the $y$ -intercept represents.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 8

Sukhila is a biologist investigating the relationship between the amount of growth hormone, $x$ grams, supplied to an orchid plant, and the number of flowers $(y)$ it produces.

$x$	$1$	$1.5$	$2$	$2.5$	$3$	$3.5$	$4$	$4.5$
$y$	$6$	$7$	$10$	$15$	$9$	$17$	$20$	$18$

Sukhila models the data using the $y$ on $x$ regression line with equation $y = a x + b$

Estimate the number of flowers that Sukhila would expect from $1.75$ grams of growth hormone.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 8

Sukhila is a biologist investigating the relationship between the amount of growth hormone, $x$ grams, supplied to an orchid plant, and the number of flowers $(y)$ it produces.

$x$	$1$	$1.5$	$2$	$2.5$	$3$	$3.5$	$4$	$4.5$
$y$	$6$	$7$	$10$	$15$	$9$	$17$	$20$	$18$

Sukhila models the data using the $y$ on $x$ regression line with equation $y = a x + b$

Sukhila wanted the plant to produce $12$ flowers. Find how much growth hormone she should use.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Chapter review>Q 8

Sukhila is a biologist investigating the relationship between the amount of growth hormone, $x$ grams, supplied to an orchid plant, and the number of flowers $(y)$ it produces.

$x$	$1$	$1.5$	$2$	$2.5$	$3$	$3.5$	$4$	$4.5$
$y$	$6$	$7$	$10$	$15$	$9$	$17$	$20$	$18$

Sukhila models the data using the $y$ on $x$ regression line with equation $y = a x + b$

Explain why it might be inappropriate to use this model to predict the number of flowers when the plant is treated with $1 kg$ of growth hormone.

The amount of fuel (y litres) used by a car to travel x km is shown in this table. Distancex km Amount of fuel (in y litres) 30 3.2 65 7.5 110 8.4 140 15.1 185 16.5 This data can be modelled by the regression line with equation y=ax+b Explain what the gradient, a, represents.

Important Questions on Modelling Relationships between Two Data Sets: Statistics for Bivariate Data

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