Correlation and Regression Questions and Answers

Q: A regression line passing through zero will have intercept value of :

The intercept is zero.A regression line has the form \(y = a + bx\), where \(a\) is the intercept, the value of \(y\) when \(x = 0\). If the line passes through the origin (0, 0), then putting \(x = 0\) gives \(y = 0\), so \(a = 0\) and the line is simply \(y = bx\).An intercept of +1, +.05 or \(-1\) would mean the line crosses the y-axis above or below the origin, so it would not pass through zero.

Q: In a partially destroyed laboratory record of an analysis of correlation data, the results only are legible as variance of X = 9 and regression equations are \(8X - 10Y + 66 = 0\) and \(40X - 18Y = 214\). What is the mean values of X and Y ?

The means are X = 13 and Y = 17.Both regression lines pass through the point of means \((\bar{X}, \bar{Y})\), so solve them together:\[ \begin{aligned} 8X - 10Y &= -66 \\ 40X - 18Y &= 214 \end{aligned} \]Multiplying the first by 5 gives \(40X - 50Y = -330\). Subtracting this from the second:\[ \begin{aligned} 32Y &= 544 \Rightarrow Y = 17 \\ 8X &= 10(17) - 66 = 104 \Rightarrow X = 13 \end{aligned} \]The variance of X is not needed for the means. 17 and 13 reverses the two values, and 12 and 17 does not satisfy the first equation.

Q: Seasonal variation means the variation occurring within :

Seasonal variation occurs within parts of a year.In time-series analysis, seasonal variation is the regular, periodic movement that repeats within a year, month by month or quarter by quarter, because of weather, festivals and customs: higher sales of woollens in winter, of umbrellas in the monsoon, or air pollution rising each winter in north India.Variation over a number of years is the secular trend or the cyclical component. Movements within a month are short-term fluctuations, not seasonal ones. 'All of the above' mixes these different components, so it is wrong.

Q: When the correlation coefficient r = 0, then the two regression lines are :

When r = 0 the regression lines are perpendicular to each other.The two regression lines are\[ \begin{aligned} y - \bar{y} &= r\frac{\sigma_{y}}{\sigma_{x}}(x - \bar{x}) \\ x - \bar{x} &= r\frac{\sigma_{x}}{\sigma_{y}}(y - \bar{y}) \end{aligned} \]With r = 0 the first becomes \(y = \bar{y}\), a horizontal line, and the second becomes \(x = \bar{x}\), a vertical line, so they cross at right angles at the point of means.The lines coincide only when r = +1 or -1, perfect correlation. They are never parallel, because both always pass through \((\bar{x}, \bar{y})\), and they exist for any data with non-zero variances.

Q: Let the value of correlation coefficient is greater than zero, the arithmetic mean of the regression coefficient is :

The arithmetic mean of the regression coefficients is greater than the correlation coefficient.The correlation coefficient is the geometric mean of the two regression coefficients, \(r = \sqrt{b_{yx} b_{xy}}\), when r is positive. By the AM-GM inequality:\[ \frac{b_{yx} + b_{xy}}{2} \ge \sqrt{b_{yx} b_{xy}} = r \]The mean equals r only in the special case \(b_{yx} = b_{xy}\), that is when the two variables have equal standard deviations, and otherwise it is greater. So it is never less than r, and 'All of the above' would include that impossible case.

Q: The equation of the linear trend fitted to the annual data is yt = 214.4 + 5.86x of an industry for the years 1998 to 2004, where x = t - 2001 and yt is the annual production in thousand tonnes in time period t. What is the estimate of the annual production for the year 2005 ?

The estimate for 2005 is 237.84 thousand tonnes.The origin is at 2001, so for 2005 \(x = 2005 - 2001 = 4\):\[ \begin{aligned} y &= 214.4 + 5.86 \times 4 \\ &= 214.4 + 23.44 = 237.84 \end{aligned} \]216.84 and 227.84 do not fit \(x = 4\), and 2017.84 is far too large to follow the trend of about 214 thousand tonnes a year.

Q: Two variables are said to be correlated if :

Two variables are correlated if a change in one variable results in a corresponding change in the other.Correlation means the two variables move together. The movement may be in the same direction (positive correlation, such as height and weight) or in opposite directions (negative correlation, such as price and demand), and its strength is measured by the correlation coefficient.Option B describes only negative correlation, so it is too narrow. Whether the variables are similar or dissimilar in nature has nothing to do with correlation; it is their changes that must correspond.

Q: Which of the following statements are correct for Karl Pearson's correlation coefficient ?(i) The scatter diagram of the data will give a straight line curve(ii) The scatter diagram of the data will not give a straight line curve(iii) Each of the variables is being affected by a large number of independent contributory causes of such a nature as to produce normal distribution(iv) The forces so operating on each of the variable series are not independent of each other but are related in a causal fashion.

The correct statements are (i), (iii) and (iv).These are the three assumptions of Karl Pearson's coefficient of correlation in the standard textbook account:Linear relationship: plotted on a scatter diagram, the paired values tend to lie along a straight line (i).Normality: each variable is affected by a large number of independent causes, so that it is normally distributed (iii).Cause and effect: the forces acting on the two series are related causally, not independent of each other (iv).(ii) contradicts (i). Pearson's r measures only the strength of linear association; when the scatter diagram shows a curve, r can be close to zero even though the variables are strongly related. Every option containing (ii) is wrong.

Q: In case of natural no. m attributes, the total number of ultimate class frequencies is :

The number of ultimate class frequencies is \(2^{m}\).In the theory of attributes, an ultimate class frequency is the frequency of a class specified by all m attributes at once, each either present or absent, such as (ABC), (AB\(\gamma\)) and so on. Each attribute gives two choices, so there are \(2 \times 2 \times \cdots = 2^{m}\) ultimate classes. With two attributes A and B they are (AB), (A\(\beta\)), (\(\alpha\)B) and (\(\alpha\beta\)), four in all.The total number of class frequencies of all orders is \(3^{m}\), since each attribute may be present, absent or not specified. \(2^{m} - 1\), \(2^{m} + 1\) and m do not count the ultimate classes.

Q: Let 8x - 10y + 66 = 0 and 40x - 18y = 214 be the lines of regression of Y on X and X on Y respectively. What is the correlation coefficient between X and Y ?

The correlation coefficient is 0.6.Write each line in its regression form. Y on X: \(10y = 8x + 66\), so \(y = 0.8x + 6.6\) and \(b_{yx} = 0.8\). X on Y: \(40x = 18y + 214\), so \(x = 0.45y + 5.35\) and \(b_{xy} = 0.45\).\[ \begin{aligned} r^{2} &= b_{yx} \times b_{xy} \\ &= 0.8 \times 0.45 = 0.36 \\ r &= 0.6 \end{aligned} \]r takes the sign of the regression coefficients, which are both positive, so r = +0.6, not -0.6. 0.5 and -0.5 do not follow from 0.36. The check \(r^{2} \le 1\) also confirms the lines were labelled correctly.

A regression line passing through zero will have intercept value of :

A.+ 1
B.+ .05
C.Zero
D.\(-\) 1

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: C

The intercept is zero.

A regression line has the form \(y = a + bx\), where \(a\) is the intercept, the value of \(y\) when \(x = 0\). If the line passes through the origin (0, 0), then putting \(x = 0\) gives \(y = 0\), so \(a = 0\) and the line is simply \(y = bx\).

An intercept of +1, +.05 or \(-1\) would mean the line crosses the y-axis above or below the origin, so it would not pass through zero.

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In a partially destroyed laboratory record of an analysis of correlation data, the results only are legible as variance of X = 9 and regression equations are \(8X - 10Y + 66 = 0\) and \(40X - 18Y = 214\). What is the mean values of X and Y ?

A.12 and 17
B.13 and 17
C.17 and 12
D.17 and 13

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: B

The means are X = 13 and Y = 17.

Both regression lines pass through the point of means \((\bar{X}, \bar{Y})\), so solve them together:

\[ \begin{aligned} 8X - 10Y &= -66 \\ 40X - 18Y &= 214 \end{aligned} \]

Multiplying the first by 5 gives \(40X - 50Y = -330\). Subtracting this from the second:

\[ \begin{aligned} 32Y &= 544 \Rightarrow Y = 17 \\ 8X &= 10(17) - 66 = 104 \Rightarrow X = 13 \end{aligned} \]

The variance of X is not needed for the means. 17 and 13 reverses the two values, and 12 and 17 does not satisfy the first equation.

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Seasonal variation means the variation occurring within :

A.A number of years
B.Parts of a year
C.Parts of a month
D.All of the above

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: B

Seasonal variation occurs within parts of a year.

In time-series analysis, seasonal variation is the regular, periodic movement that repeats within a year, month by month or quarter by quarter, because of weather, festivals and customs: higher sales of woollens in winter, of umbrellas in the monsoon, or air pollution rising each winter in north India.

Variation over a number of years is the secular trend or the cyclical component. Movements within a month are short-term fluctuations, not seasonal ones. 'All of the above' mixes these different components, so it is wrong.

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When the correlation coefficient r = 0, then the two regression lines are :

A.Perpendicular to each other
B.Parallel to each other
C.Coincide
D.Do not exist

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

When r = 0 the regression lines are perpendicular to each other.

The two regression lines are

\[ \begin{aligned} y - \bar{y} &= r\frac{\sigma_{y}}{\sigma_{x}}(x - \bar{x}) \\ x - \bar{x} &= r\frac{\sigma_{x}}{\sigma_{y}}(y - \bar{y}) \end{aligned} \]

With r = 0 the first becomes \(y = \bar{y}\), a horizontal line, and the second becomes \(x = \bar{x}\), a vertical line, so they cross at right angles at the point of means.

The lines coincide only when r = +1 or -1, perfect correlation. They are never parallel, because both always pass through \((\bar{x}, \bar{y})\), and they exist for any data with non-zero variances.

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Let the value of correlation coefficient is greater than zero, the arithmetic mean of the regression coefficient is :

A.Greater than the correlation coefficient
B.Less than the correlation coefficient
C.Equal to the correlation coefficient
D.All of the above

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: A

The arithmetic mean of the regression coefficients is greater than the correlation coefficient.

The correlation coefficient is the geometric mean of the two regression coefficients, \(r = \sqrt{b_{yx} b_{xy}}\), when r is positive. By the AM-GM inequality:

\[ \frac{b_{yx} + b_{xy}}{2} \ge \sqrt{b_{yx} b_{xy}} = r \]

The mean equals r only in the special case \(b_{yx} = b_{xy}\), that is when the two variables have equal standard deviations, and otherwise it is greater. So it is never less than r, and 'All of the above' would include that impossible case.

Discuss in Forum

The equation of the linear trend fitted to the annual data is y_t = 214.4 + 5.86x of an industry for the years 1998 to 2004, where x = t - 2001 and y_t is the annual production in thousand tonnes in time period t. What is the estimate of the annual production for the year 2005 ?

A.216.84 thousand tonnes
B.2017.84 thousand tonnes
C.227.84 thousand tonnes
D.237.84 thousand tonnes

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: D

The estimate for 2005 is 237.84 thousand tonnes.

The origin is at 2001, so for 2005 \(x = 2005 - 2001 = 4\):

\[ \begin{aligned} y &= 214.4 + 5.86 \times 4 \\ &= 214.4 + 23.44 = 237.84 \end{aligned} \]

216.84 and 227.84 do not fit \(x = 4\), and 2017.84 is far too large to follow the trend of about 214 thousand tonnes a year.

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Two variables are said to be correlated if :

A.The change in one variable results in a corresponding change in other variable
B.The change of the variables is in the opposite directions
C.The two variables are similar in nature
D.The two variables are dissimilar in nature

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: A

Two variables are correlated if a change in one variable results in a corresponding change in the other.

Correlation means the two variables move together. The movement may be in the same direction (positive correlation, such as height and weight) or in opposite directions (negative correlation, such as price and demand), and its strength is measured by the correlation coefficient.

Option B describes only negative correlation, so it is too narrow. Whether the variables are similar or dissimilar in nature has nothing to do with correlation; it is their changes that must correspond.

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Which of the following statements are correct for Karl Pearson's correlation coefficient ?

(i) The scatter diagram of the data will give a straight line curve
(ii) The scatter diagram of the data will not give a straight line curve
(iii) Each of the variables is being affected by a large number of independent contributory causes of such a nature as to produce normal distribution
(iv) The forces so operating on each of the variable series are not independent of each other but are related in a causal fashion.

A.(i), (ii) and (iii)
B.(i), (ii) and (iv)
C.(i), (iii) and (iv)
D.(ii), (iii) and (iv)

Exam Year(s)

MH SET - 2025, SET Exam - 2025,

Answer: C

The correct statements are (i), (iii) and (iv).

These are the three assumptions of Karl Pearson's coefficient of correlation in the standard textbook account:

Linear relationship: plotted on a scatter diagram, the paired values tend to lie along a straight line (i).
Normality: each variable is affected by a large number of independent causes, so that it is normally distributed (iii).
Cause and effect: the forces acting on the two series are related causally, not independent of each other (iv).

(ii) contradicts (i). Pearson's r measures only the strength of linear association; when the scatter diagram shows a curve, r can be close to zero even though the variables are strongly related. Every option containing (ii) is wrong.

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In case of natural no. m attributes, the total number of ultimate class frequencies is :

A.\(2^{m}\)
B.\(2^{m}-1\)
C.\(2^{m}+1\)
D.m

Exam Year(s)

MH SET - 2025, SET Exam - 2025,

Answer: A

The number of ultimate class frequencies is \(2^{m}\).

In the theory of attributes, an ultimate class frequency is the frequency of a class specified by all m attributes at once, each either present or absent, such as (ABC), (AB\(\gamma\)) and so on. Each attribute gives two choices, so there are \(2 \times 2 \times \cdots = 2^{m}\) ultimate classes. With two attributes A and B they are (AB), (A\(\beta\)), (\(\alpha\)B) and (\(\alpha\beta\)), four in all.

The total number of class frequencies of all orders is \(3^{m}\), since each attribute may be present, absent or not specified. \(2^{m} - 1\), \(2^{m} + 1\) and m do not count the ultimate classes.

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10.

Let 8x - 10y + 66 = 0 and 40x - 18y = 214 be the lines of regression of Y on X and X on Y respectively. What is the correlation coefficient between X and Y ?

A.0.5
B.- 0.5
C.0.6
D.- 0.6

Exam Year(s)

MH SET - 2025, SET Exam - 2025,

Answer: C

The correlation coefficient is 0.6.

Write each line in its regression form. Y on X: \(10y = 8x + 66\), so \(y = 0.8x + 6.6\) and \(b_{yx} = 0.8\). X on Y: \(40x = 18y + 214\), so \(x = 0.45y + 5.35\) and \(b_{xy} = 0.45\).

\[ \begin{aligned} r^{2} &= b_{yx} \times b_{xy} \\ &= 0.8 \times 0.45 = 0.36 \\ r &= 0.6 \end{aligned} \]

r takes the sign of the regression coefficients, which are both positive, so r = +0.6, not -0.6. 0.5 and -0.5 do not follow from 0.36. The check \(r^{2} \le 1\) also confirms the lines were labelled correctly.

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