Morgan records the length and height, in $\mathrm{cm}$, of each of seven cats. Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 Find the value of $r$ for this data. Interpret the $r$ -value in this context.

Given that, Morgan records the length and height, in cm , of each of seven cats. Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 From the given, x ¯ = ∑ x n = 305 7 = 43 . 57 y ¯ = ∑ y n = 196 7 = 28 x 2 = 35 2 + 38 2 + 62 2 + 45 2 + 47 2 + 48 2 + 50 2 = 13471 Serial number length x height y x y X = x - x Y = y - y X Y 1 35 13 455 - 8 . 57 - 15 128 . 55 2 38 16 684 - 5 . 57 - 10 55 . 7 3 42 27 1134 - 1 . 57 - 1 1 . 57 4 45 28 1260 1 . 43 0 0 5 47 36 1692 3 . 43 8 27 . 44 6 48 34 1632 4 . 43 6 26 . 58 7 50 40 2000 6 . 43 12 77 . 16 Total ∑ x = 305 ∑ y = 196 ∑ x y = 8857 ∑ X Y = 317 From table, we can find that, ∑ X 2 = - 8 . 87 2 + - 5 . 57 2 + - 1 . 57 2 + 1 . 43 2 + 3 . 43 2 + 4 . 43 2 + 6 . 43 2 = 181 . 714 ∑ Y 2 = - 15 2 + - 10 2 + - 1 2 + 0 2 + 8 2 + 6 2 + 12 2 = 570 ⇒ r = ∑ x y ∑ x 2 ∑ y 2 = 317 181 . 714 × 570 = 317 23 . 87 × 13 . 47 = 0 . 98 The value of r = 0 . 98 i.e. 0 . 75 < r ⩽ 1 ∴ Perfect positive correlation between length and height.

Morgan records the length and height, in cm , of each of seven cats. Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 Find the equation of the y on x regression line.

Given data: Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 Here, x 2 = 35 2 + 38 2 + 62 2 + 45 2 + 47 2 + 48 2 + 50 2 = 13471 Serial number length x height y x y 1 35 13 455 2 38 16 684 3 42 27 1134 4 45 28 1260 5 47 36 1692 6 48 34 1632 7 50 40 2000 Total ∑ x = 305 ∑ y = 196 ∑ x y = 8857 The line of regression of y on x be, y = b + a x Now, ∑ y = n b + a ∑ x ⇒ 196 = 7 b + 305 a and ∑ x y = b ∑ x + a ∑ x 2 ⇒ 8857 = 305 b + 13471 a Solving the both equation we get, a = 1 . 744 and b = - 48 . 010 Thus, the line of regression is y = - 48 . 010 + 1 . 744 x

Morgan records the length and height, in cm , of each of seven cats. Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 Morgan's new cat has a length of 40 cm , but it would not stay still for long enough for its height to be measured.

Given data: Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 Here, x 2 = 35 2 + 38 2 + 62 2 + 45 2 + 47 2 + 48 2 + 50 2 = 13471 Serial number length x height y x y 1 35 13 455 2 38 16 684 3 42 27 1134 4 45 28 1260 5 47 36 1692 6 48 34 1632 7 50 40 2000 Total ∑ x = 305 ∑ y = 196 ∑ x y = 8857 The line of regression of y on x be, y = b + a x Now, ∑ y = n b + a ∑ x ⇒ 196 = 7 b + 305 a and ∑ x y = b ∑ x + a ∑ x 2 ⇒ 8857 = 305 b + 13471 a Solving the both equation we get, a = 1 . 744 and b = - 48 . 010 Thus, the line of regression is y = - 48 . 010 + 1 . 744 x put x = 40 cm , y = - 48 . 010 + 1 . 744 × 40 ⇒ y = 21 . 75 Morgan's new car has a length of 40 cm , but it would not stay still for long enough for its height to be measured is 21 . 75 cm .

Morgan records the length and height, in $\mathrm{cm}$, of each of seven cats. Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 interpret the meaning of the gradient in the equation of this regression line.

Given data: Length cm 35 38 42 45 47 48 50 Height cm 13 18 27 28 36 34 40 Here, x 2 = 35 2 + 38 2 + 62 2 + 45 2 + 47 2 + 48 2 + 50 2 = 13471 Serial number length x height y x y 1 35 13 455 2 38 16 684 3 42 27 1134 4 45 28 1260 5 47 36 1692 6 48 34 1632 7 50 40 2000 Total ∑ x = 305 ∑ y = 196 ∑ x y = 8857 The line of regression of y on x be, y = b + a x Now, ∑ y = n b + a ∑ x ⇒ 196 = 7 b + 305 a and ∑ x y = b ∑ x + a ∑ x 2 ⇒ 8857 = 305 b + 13471 a Solving the both equation we get, a = 1 . 744 and b = - 48 . 010 Thus, the line of regression is y = - 48 . 010 + 1 . 744 x . From regression equation, 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 Thus, for every cm that the cat grows in length, it grows 1 . 74 cm in height.

At low tide, mud banks are exposed on both sides of a river. Sonny recorded the number of mudbugs at various distances from the river's edge. Distance m 0 . 5 1 1 . 5 2 2 . 5 3 4 Number of mudbugs 30 28 14 18 10 7 1 Sonny uses his GDC to model the relationship between the variables by a regression equation of the form y = a x + b . Write down the values of a and b .

Given data: Distance m 0 . 5 1 1 . 5 2 2 . 5 3 4 Number of mudbugs 30 28 14 18 10 7 1 From the table, ∑ x 2 = 0 . 5 2 + 1 2 + 1 . 5 2 + 2 2 + 2 . 5 2 + 3 2 + 4 2 = 38 . 75 Serial number Distance x Number of mudbugs y x y 1 0 . 5 30 15 2 1 28 28 3 1 . 5 14 21 4 2 18 96 5 2 . 5 10 25 6 3 7 21 7 3 . 5 1 4 . Total ∑ x = 14 . 5 ∑ y = 108 ∑ x y = 150 The lie of regression of y on x be y = a + b x Now, ∑ y = n b + a ∑ x ⇒ 108 = 7 b + 14 . 5 a ...(i) ∑ x y = b ∑ x + a ∑ x 2 ⇒ 150 = 14 . 5 b + 38 . 75 a ...........(ii) Solving equation (i) and (ii) we get, a = - 8 . 459 and b = 32 . 951 . Thus, the line of regression is y = - 8 . 459 x + 32 . 951 .

EASY

Diploma

IMPORTANT

Earn 100

Ms Fung's class has $10$ students. Their scores for both classwork and the final exam are shown below. Both scores are out of $100$ .

Student Lin Jung Erika Non Chen Park Ploy Jai Fae Nit

Classwork $80$ $73$ $95$ $84$ $67$ $88$ $69$ $92$ $75$ $90$

Final $74$ $62$ $93$ $75$ $73$ $81$ $58$ $90$ Abs $84$

Find the equation of the $y$ on $x$ regression line.

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>Exercise 7D>Q 5

Ms Fung's class has $10$ students. Their scores for both classwork and the final exam are shown below. Both scores are out of $100$ .

Student	Lin	Jung	Erika	Non	Chen	Park	Ploy	Jai	Fae	Nit
Classwork	$80$	$73$	$95$	$84$	$67$	$88$	$69$	$92$	$90$	$84$
Final	$74$	$62$	$738$	$690$	$57642$	$61368$	$58$	$90$	Abs	$84$

Use the equation of the regression line to estimate Fae's score, had she been present for the final exam.

MEDIUM

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Exercise 7D>Q 6

Morgan records the length and height, in $\mathrm{cm}$, of each of seven cats.

Length $(cm)$	$35$	$38$	$42$	$45$	$47$	$48$	$50$
Height $(cm)$	$13$	$18$	$27$	$28$	$36$	$34$	$40$

Find the value of $r$ for this data. Interpret the $r$ -value in this context.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Exercise 7D>Q 6

Morgan records the length and height, in $cm$ , of each of seven cats.

Length $(cm)$	$35$	$38$	$42$	$45$	$47$	$48$	$50$
Height $(cm)$	$13$	$18$	$27$	$28$	$36$	$34$	$40$

Find the equation of the $y$ on $x$ regression line.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Exercise 7D>Q 6

Morgan records the length and height, in $cm$ , of each of seven cats.

Length $(cm)$	$35$	$38$	$42$	$45$	$47$	$48$	$50$
Height $(cm)$	$13$	$18$	$27$	$28$	$36$	$34$	$40$

Morgan's new cat has a length of $40 cm$ , but it would not stay still for long enough for its height to be measured.

EASY

Diploma

IMPORTANT

Mathematics : Analysis and Approaches Standard Level Course Companion>Chapter 7 - Modelling Relationships between Two Data Sets: Statistics for Bivariate Data>Exercise 7D>Q 6

Morgan records the length and height, in $\mathrm{cm}$, of each of seven cats.

Length $(cm)$	$35$	$38$	$42$	$45$	$47$	$48$	$50$
Height $(cm)$	$13$	$18$	$27$	$28$	$36$	$34$	$40$

interpret the meaning of the gradient in the equation of this regression line.

EASY

Diploma

IMPORTANT

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

At low tide, mud banks are exposed on both sides of a river. Sonny recorded the number of mudbugs at various distances from the river's edge.

Distance $(m)$	$0.5$	$1$	$1.5$	$2$	$2.5$	$3$	$4$
Number of mudbugs	$30$	$28$	$14$	$18$	$10$	$7$	$1$

Sonny uses his GDC to model the relationship between the variables by a regression equation of the form $y = a x + b$ .
Write down the values of $a and b$ .

EASY

Diploma

IMPORTANT

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

At low tide, mud banks are exposed on both sides of a river. Sonny recorded the number of mudbugs at various distances from the river's edge.

Distance $(m)$	$0.5$	$1$	$1.5$	$2$	$2.5$	$3$	$4$
Number of mudbugs	$30$	$28$	$14$	$18$	$10$	$7$	$1$

Sonny uses his GDC to model the relationship between the variables by a regression equation of the form $y = a x + b$ .

Hence, estimate the number of mudbugs, to the nearest whole number, that are

3.5 m

from the river's edge.

EASY

Diploma

IMPORTANT

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

At low tide, mud banks are exposed on both sides of a river. Sonny recorded the number of mudbugs at various distances from the river's edge.

Distance $(m)$	$0.5$	$1$	$1.5$	$2$	$2.5$	$3$	$4$
Number of mudbugs	$30$	$28$	$14$	$18$	$10$	$7$	$1$

Find Pearson's product-moment correlation coefficient $(r)$ for this data.[Answer up to two decial values]

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

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