HSC Commerce (Marathi Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

The equations of the two lines of regression are 6x + y − 31 = 0 and 3x + 2y – 26 = 0. Calculate the mean values of x and y

Two samples from bivariate populations have 15 observations each. The sample means of X and Y are 25 and 18 respectively. The corresponding sum of squares of deviations from means are 136 and 148 respectively. The sum of product of deviations from respective means is 122. Obtain the regression equation of x on y

If n = 5, Σx = Σy = 20, Σx² = Σy² = 90, Σxy = 76 Find the regression equation of x on y

If n = 6, Σx = 36, Σy = 60, Σxy = –67, Σx² = 50, Σy² =106, Estimate y when x is 13

The regression equation of x on y is 40x – 18y = 214  ......(i)

The regression equation of y on x is 8x – 10y + 66 = 0  ......(ii)

Solving equations (i) and (ii),

`barx = square`

`bary = square`

∴ b_yx = `square/square`

∴ b_xy = `square/square`

∴ r = `square`

Given variance of x = 9

∴ b_yx = `square/square`

∴ `sigma_y = square`

The complicated but efficient method of measuring trend of time series is ______

The simplest method of measuring trend of time series is ______

The method of measuring trend of time series using only averages is ______

State whether the following statement is True or False:

The secular trend component of time series represents irregular variations

State whether the following statement is True or False: 

Moving average method of finding trend is very complicated and involves several calculations

State whether the following statement is True or False:

Least squares method of finding trend is very simple and does not involve any calculations

Following table shows the amount of sugar production (in lac tons) for the years 1971 to 1982

Fit a trend line by the method of least squares

Obtain trend values for data, using 4-yearly centred moving averages

The following table gives the production of steel (in millions of tons) for years 1976 to 1986.

Obtain the trend value for the year 1990

Obtain the trend values for the data, using 3-yearly moving averages

Use the method of least squares to fit a trend line to the data given below. Also, obtain the trend value for the year 1975.

The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

1. Obtain trend values for the above data using 5-yearly moving averages.
2. Plot the original time series and trend values obtained above on the same graph.

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010

Fit a trend line by the method of least squares

Solution: Let us fit equation of trend line for above data.

Let the equation of trend line be y = a + bx   .....(i)

Here n = 7(odd), middle year is `square` and h = 5

The normal equations are

Σy = na + bΣx

As, Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx²

As, Σx = 0, b =`square`

∴ The equation of trend line is y = `square`

Obtain trend values for data, using 3-yearly moving averages
Solution:

Fit equation of trend line for the data given below.

Let the equation of trend line be y = a + bx   .....(i)

Here n = `square` (even), two middle years are `square` and 2011, and h = `square`

The normal equations are Σy = na + bΣx

As Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx²

As Σx = 0, b = `square`

Substitute values of a and b in equation (i) the equation of trend line is `square`

To find trend value for the year 2016, put x = `square` in the above equation.

y = `square`