The following table shows the ages x and systolic blood pressure y of 12 women. Determine the least squares regression line of y on x . Age ( x ) 56 42 72 36 63 47 55 49 38 42 68 60 Blood Pressure ( y ) 147 125 160 118 149 128 150 145 115 140 152 155

Given data can be tabulated as x y x - x y - y ( x - x ) 2 ( y - y ) 2 ( x - x ) ( y - y ) 56 147 3 . 67 6 . 67 13 . 5 44.5 24.5 42 125 - 10 . 33 - 15 . 33 106 . 7 235 158 . 36 72 160 19 . 67 19 . 67 386 . 9 386 . 9 386 . 9 36 118 - 16 . 33 - 22 . 33 266 . 7 498 . 6 364 . 64 63 149 10 . 67 8 . 67 113 . 85 75 . 2 92 . 5 47 128 - 5 . 33 - 12 . 33 28 . 4 152 65 . 7 55 150 2 . 67 9 . 67 7 . 1 93 . 5 25 . 82 49 145 - 3 . 33 4 . 67 11 21 . 8 - 15 . 55 38 115 - 14 . 33 - 25 . 33 205 . 35 641 . 6 363 42 140 - 10 . 33 - 0 . 33 106 . 7 0 . 1 3 . 4 68 152 15 . 67 11 . 67 245 . 5 136 . 2 183 60 155 7 . 67 14 . 67 58 . 8 215 . 2 112 . 5 ∑ x = 628 ∑ y = 1684 ∑ = 0 ∑ = 0 ∑ = 1551 ∑ = 2500 . 6 ∑ = 1765 Mean of x , x = ∑ x n = 628 12 = 52 . 33 Mean of y , y = ∑ y n = 1684 12 = 140 . 33 b y x = ∑ ( x - x ) ( y - y ) ∑ ( x - x ) 2 b y x = 1765 1551 = 1 . 14 Regression line of y on x is y - y = b y x ( x - x ) y - 140 . 33 = 1 . 14 ( x - 52 . 33 ) y = 1 . 14 x + 80 . 67

The following table show final grades of 10 students in Algebra and Physics. Algebra ( x ) 75 80 93 65 87 71 98 68 84 77 Physics ( y ) 82 78 86 72 91 80 95 72 89 74 Find both the least squares lines.

Let assumed mean A = 80 , B = 82 Given data can be tabulated as x y u = x - A v = y - B u 2 v 2 u v 75 82 - 5 0 25 0 0 80 78 0 - 4 0 16 0 93 86 13 4 169 16 52 65 72 - 15 - 10 225 100 150 87 91 7 9 49 81 63 71 80 - 9 - 2 81 4 18 98 95 18 13 324 169 234 68 72 - 12 - 10 144 100 120 84 89 4 7 16 49 28 77 74 - 3 - 8 9 64 24 ∑ x = 798 ∑ y = 819 ∑ u = - 2 ∑ v = - 1 ∑ u 2 = 1042 ∑ v 2 = 599 ∑ u v = 689 Mean of x , x = A + ∑ u n = 80 + - 2 10 = 79 . 8 Mean of y , y = B + ∑ v n = 82 + - 1 10 = 81 . 9 b y x = ∑ u v - 1 n ∑ u ∑ v ∑ u 2 - 1 n ∑ u 2 , b x y = ∑ u v - 1 n ∑ u ∑ v ∑ v 2 - 1 n ∑ v 2 b y x = 689 - 0 . 1 ( - 2 ) ( - 1 ) 1042 - 0 . 1 ( - 2 ) ( - 2 ) = 688 . 8 1041 . 6 = 0 . 66 b x y = 689 - 0 . 1 ( - 2 ) ( - 1 ) 599 - 0 . 1 ( - 1 ) ( - 1 ) = 688 . 8 598 . 9 = 1 . 15 Regression line of y on x is y - y = b y x ( x - x ) y - 81 . 9 = 0 . 66 ( x - 79 . 8 ) y = 0 . 66 x + 29 . 23 Regression line of x on y is x - x = b x y ( y - y ) ( x - 79 . 8 ) = 1 . 15 ( y - 81 . 9 ) x = 1 . 15 y - 14 . 4

For observations of pairs ( x , y ) of the variables x and y , the results were obtained : ∑ x = 125 , ∑ y = 100 , ∑ x 2 = 1650 , ∑ y 2 = 1500 , ∑ x y = 50 and n = 25 . Find the equation of the line of regression of x on y . Hence, estimate the value of x if y = 5 .

Line of regression of x on y is given by, x - x = b x y ( y - y ) Now, y = ∑ y n = 100 25 = 4 , x = ∑ x n = 125 25 = 5 b x y = ∑ x y - 1 n ∑ x ∑ y ∑ y 2 - ( ∑ y ) 2 b x y = 50 - 1 25 ( 125 ) ( 100 ) 1500 - 1 25 ( 100 ) 2 = - 450 1100 = - 9 22 Line of regression, x - 5 = - 9 22 ( y - 4 ) 22 x + 9 y = 146 Substituting y = 5 22 x + 9 ( 5 ) = 146 x = 4 . 59

HARD

12th ICSE

IMPORTANT

Earn 100

The following table shows the ages $x$ and systolic blood pressure $y$ of $12$ women.

Find the correlation coefficient between $x$ and $y$ . (Answer upto two places of decimal)

Age ( $x$ ) $56$ $42$ $72$ $36$ $63$ $47$ $55$ $49$ $38$ $42$ $68$ $60$

Blood Pressure ( $y$ ) $147$ $125$ $160$ $118$ $149$ $128$ $150$ $145$ $115$ $140$ $152$ $155$

50% studentsanswered this correctly

Important Questions on Linear Regression

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>EXERCISE>Q 21

The following table shows the ages $x$ and systolic blood pressure $y$ of $12$ women.

Determine the least squares regression line of $y$ on $x$ .

Age ( $x$ )	$56$	$42$	$72$	$36$	$63$	$47$	$55$	$49$	$38$	$42$	$68$	$60$
Blood Pressure ( $y$ )	$147$	$125$	$160$	$118$	$149$	$128$	$150$	$145$	$115$	$140$	$152$	$155$

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>EXERCISE>Q 21

The following table shows the ages $x$ and systolic blood pressure $y$ of $12$ women.

Estimate the blood pressure of a woman whose age is $45$ years.

Age ( $x$ )	$56$	$42$	$72$	$36$	$63$	$47$	$55$	$49$	$38$	$42$	$68$	$60$
Blood Pressure ( $y$ )	$147$	$125$	$160$	$118$	$149$	$128$	$150$	$145$	$115$	$140$	$152$	$155$

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>EXERCISE>Q 22

The following table show final grades of $10$ students in Algebra and Physics.

Algebra ( $x$ )	$75$	$80$	$93$	$65$	$87$	$71$	$98$	$68$	$84$	$77$
Physics ( $y$ )	$82$	$78$	$86$	$72$	$91$	$80$	$95$	$72$	$89$	$74$

Find both the least squares lines.

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>EXERCISE>Q 22

The following table show final grades of $10$ students in Algebra and Physics.

Algebra (x)	$75$	$80$	$93$	$65$	$87$	$71$	$98$	$68$	$84$	$77$
Physics (y)	$82$	$78$	$86$	$72$	$91$	$80$	$95$	$72$	$89$	$74$

If a student receives

75

marks in Algebra, what is his expected score in Physics ?

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>EXERCISE>Q 22

The following table show final grades of $10$ students in Algebra and Physics.

Algebra ( $x$ )	$75$	$80$	$93$	$65$	$87$	$71$	$98$	$68$	$84$	$77$
Physics ( $y$ )	$82$	$78$	$86$	$72$	$91$	$80$	$95$	$72$	$89$	$74$

If a student receives

95

marks in Physics, what is his expected score in Algebra?

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>CHAPTER TEST>Q 1

If $x$ and $y$ are a pair of correlated variables. Ten observations of their values $(x_{i}, y_{i})$ have the following results :

$\sum_{}^{} x_{i} = 55, \sum_{}^{} y_{i} = 55, \sum_{}^{} x_{i} y_{i} = 350 a n d \sum_{}^{} {x_{i}}^{2} = 385 .$

Predict the value of $y$ when the value of $x$ is $6 .$

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>CHAPTER TEST>Q 2

For observations of pairs $(x, y)$ of the variables $x$ and $y$ , the results were obtained :

$\sum_{}^{} x = 125, \sum_{}^{} y = 100, \sum_{}^{} x^{2} = 1650, \sum_{}^{} y^{2} = 1500, \sum_{}^{} x y = 50 and n = 25 .$

Find the equation of the line of regression of $x$ on $y$ . Hence, estimate the value of $x$ if $y = 5 .$

EASY

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 2 - Linear Regression>CHAPTER TEST>Q 3

The coefficient of correlation between the ages of husbands and wives in a community was found to be $0.8,$ the mean of husband's age was $25$ years and that of wives $22$ years. Their standard deviations were $4$ and $5$ years respectively. Find the two lines of regression.

The following table shows the ages x and systolic blood pressure y of 12 women. Find the correlation coefficient between x and y. (Answer upto two places of decimal) Age (x) 56 42 72 36 63 47 55 49 38 42 68 60 Blood Pressure (y) 147 125 160 118 149 128 150 145 115 140 152 155

Important Questions on Linear Regression

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