Which of the following are normal equations to fit a straight line trend y = a + bx by the method of least squares? Option 1: y = a + x + b x^2, xy = a + x + b x^2 Option 2: y = na + b x, xy = a x + b x^2 Option 3: xy = na + b x, xy = na + b x^2 Option 4: x = na + b y^2, y = a x + b y^2

Answer: y = na + b x, xy = a x + b x^2. Solution: For a straight line y = a + bx fitted to data points using the method of least squares, the normal equations are derived to find the optimal values of a and b. Taking the line equation y = a + bx and summing over all n data points: y = (a + bx) y = a + bx y = na + b x Multiplying both sides of y = a + bx by x: xy = ax + bx^2 Summing over all data points: xy = (ax + bx^2) xy = a x + b x^2 The two normal equations are: y = na + b x xy = a x + b x^2 Checking the options: Option 1: First equation has a + x instead of na, which is incorrect. Option 3: Both equations start with xy, which is incorrect. Option 4: Uses x instead of y in the first equation, which is incorrect. Option 2 matches the derived normal equations exactly. Therefore, the answer is Option 2.

Statistics & Applications

Trends & Data

Medium

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Which of the following are normal equations to fit a straight line trend y = a + bx by the method of least squares?

$\sum y = a + \sum x + b \sum x^2$ , $\sum xy = a + \sum x + b \sum x^2$

$\sum y = na + b \sum x$ , $\sum xy = a \sum x + b \sum x^2$

$\sum xy = na + b \sum x$ , $\sum xy = na + b \sum x^2$

$\sum x = na + b \sum y^2$ , $\sum y = a \sum x + b \sum y^2$

✅ Correct Option: 2

For a straight line $y = a + bx$ fitted to data points using the method of least squares, the normal equations are derived to find the optimal values of $a$ and $b$ .

Taking the line equation $y = a + bx$ and summing over all $n$ data points:

$\sum y = \sum(a + bx)$

$\sum y = \sum a + \sum bx$

$\sum y = na + b\sum x$

Multiplying both sides of $y = a + bx$ by $x$ :

$xy = ax + bx^2$

Summing over all data points:

$\sum xy = \sum(ax + bx^2)$

$\sum xy = a\sum x + b\sum x^2$

The two normal equations are:

$\sum y = na + b\sum x$

$\sum xy = a\sum x + b\sum x^2$

Checking the options:

Option 1: First equation has $a + \sum x$ instead of $na$ , which is incorrect.

Option 3: Both equations start with $\sum xy$ , which is incorrect.

Option 4: Uses $\sum x$ instead of $\sum y$ in the first equation, which is incorrect.

Option 2 matches the derived normal equations exactly.

Therefore, the answer is Option 2.

2026: 22 May Shift 1

For the given 9 values, 4,5,5,6,7,8,9,8,10, the three years moving averages are

2025: 21 May Shift 2

For the given five values, 13, 17, 21, 22, 32; the 3-year moving averages are:

2025: 13 May Shift 1

Which of the following statements are correct?

(A) The method of least squares determines the position of the trend line of the given time series.

(B) The trend line is called the line of best fit.

(C) The line of best fit is a line in which the sum of deviations of the actual values of the variable from their corresponding trend value is always positive.

(D) The normal equations of the trend line $y = a + bx$ are $\sum y = na + b \sum x$ and $\sum xy = a \sum x + b \sum x^2$ , where $n$ is the numbers of observations.

Choose the correct answer from the options given below:

2025: 21 May Shift 2

Let $y = 138.86 + 7.64(x - 2021)$ be a straight line of best fit by using least square method to the following data:

Year(x)	2018	2019	2020	2021	2022	2023	2024
Profit(y) (in Rs. '000)	114	130	126	144	138	156	164

Then the trend value for the year 2024 is:

2025: 13 May Shift 1

If $y = 38.85 + 7.64(x - 2020)$ is the equation of the straight line trend for the following data:

Year $(x)$	2017	2018	2019	2020	2021	2022	2023
Profit (Rs. '000) $(y)$	14	30	26	44	38	56	64

The trend value for the year 2021 by least square method is:

2026: 12 May Shift 2

Match the components of time series with the examples associated to them.

Match List-I with List-II

List-I	List-II
(A) Secular Trend	(I) Monthly sale of soft drink increases in summer
(B) Seasonal variations	(II) Rise and fall of the share market
(C) Cyclic Variations	(III) Data regarding national income
(D) Irregular Variations	(IV) Sudden fall in crop yield due to floods

Choose the correct answer from the options given below:

Which of the following are normal equations to fit a straight line trend y = a + bx by the method of least squares?

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