Q: The standard deviation of 5 scores 1,2,3,4,5 is a . The value of ' a ' is

aligned & Mean ( x )= 1+2+3+4+5 5 =3 \\ & S.D = = 1 5 (1+4+9+16+25)-9 = 11-9 \\ & = 2 aligned

Question 1

If the variance of the data 2,3,5,8,12 is ^2 and the mean deviation from the median for this data is M , then ^2- M =

Accepted Answer

Observations : 2,3,5,8,12 Mean = 2+3+5+8+12 5 =6 ^2=13.2 Median =5=m Mean deviation about median |x_i-m | n = 3+2+0+3+7 5 =3 M=3 ^2-M=13.2-3=10.2

Question 2

The mean of n items is X . If the first item is increased by 1 , second by 2 and so on, the new mean is :

Accepted Answer

Let the items be a _1, a _2, . ., a _ n then X = a_1+a_2+ .+a_n n Now, according to the given condition: X = (a_1+1 )+ (a_2+2 )+ + (a_n+n ) n = X + 1+2+3+ +n n = X + n(n+1) 2 n (using sum of n natural nos.) = X + n +1 2 .

Question 3

The variance of 20 observations is 5 . If each observation is multiplied by 2 , then the new variance of the resulting observation is

Accepted Answer

Let the observations be x_1, x_2, , x_ 20 and x be their mean. Given that, variance =5 and n= 20. We know that, Variance ( ^2 )= 1 n _ i=1 ^ 20 (x_i- x )^2 i.e. 5= 1 20 _ i=1 ^ 20 (x_i- x )^2 or _ i=1 ^ 20 (x_i- x )^2=100 .....(i) If each observation is multiplied by 2 and the new resulting observations are y_i, then y_i=2 x_i i.e., x_i= 1 2 y_i Therefore, y = 1 n _ i =1 ^ 20 y _ i = 1 20 _ i =1 ^ 20 2 x _ i =2 1 20 _ i =1 ^ 20 x _ i i.e., y =2 x or x = 1 2 y . On substituting the values of x_i

Question 4

If the function f(x), defined below, is continuous on the interval [0,8], then f(x)= \ array cc x^2+a x+b & , 0 x 2 \ 3 x+2, & 2 x 4 \ 2 a x+5 b, & 4 x 8 array .

Accepted Answer

Here, f(x) is continuous on the interval [0,8]. So it will also be continuous on 2 and 4. At x=2, _ x 2^ - f(x)= _ x 2^ + f(x) _ x 2^ - (x^2+ ax + b )= _ x 2^ + (3 x+2) 4+2 a+b=3(2)+2 2 a+b=4 ...(i) At x=4 _ x 4^ - f(x)= _ x 4^ + f(x) _ x 4^ - (3 x+2)= _ x 4^ + 2 a x+5 b 3(4)+2=2 a(4)+5 b 8 a+5 b=14 .......(ii) On solving Eqs. (i) and (ii), we get a=3 and b=-2.

Question 5

The standard deviation of 5 scores 1,2,3,4,5 is a . The value of ' a ' is

Accepted Answer

aligned & Mean ( x )= 1+2+3+4+5 5 =3 \ & S.D = = 1 5 (1+4+9+16+25)-9 = 11-9 \ & = 2 aligned

Question 6

Find the mean deviation about the mean for the data 4,7,8,9,10,12,13,17

Accepted Answer

Arithmetic mean x of 4,7,8,9,10,12,13, 17 is x = 4+7+8+9+10+12+13+17 8 = 80 8 =10 |x_i- x |=6+3+2+1+0+2+3+7=24 Mean deviation about mean = M.D. ( x )= |x_i- x | n = 24 8 =3

Question 7

The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1,2 and 6 , then the other two are

Accepted Answer

Let the two unknown items be x and y. Then, Mean =4 1+2+6+x+y 5 =4 x+y=11 (i) and variance =5.2 1^2+2^2+6^2+x^2+y^2 5 -( mean )^2=5.2 41+x^2+y^2=5(5.2+16) array ll & 41+x^2+y^2=106 \ & x^2+y^2=65 array On solving Eqs. (i) and (ii), we get x=4, y=7 or x=7, y=4

Statistics — Previous Year Questions & Solutions