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.

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.

Discuss in Forum

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