Solveeit Logo
Login

Question

Mathematics Question on Statistics

Let the mean and the variance of 6 observation a,b, 68, 44, 48, 60 be 55 and 194, respectively if a > b, then a + 3b is

A

200

B

190

C

180

D

210

Answer

180

Explanation

Solution

Set up the equation for the mean. The mean of the six observations is given as 55. So,

a+b+68+44+48+606=55.\frac{a + b + 68 + 44 + 48 + 60}{6} = 55.

Multiply both sides by 6 to eliminate the denominator:

a+b+68+44+48+60=330.a + b + 68 + 44 + 48 + 60 = 330.

Simplify to get:

a+b=110(Equation 1).a + b = 110 \quad \text{(Equation 1)}.

Set up the equation for the variance. The variance of the six observations is given as 194.

Recall that the variance formula for a set of observations x1,x2,…,xnx_1, x_2, \ldots, x_n with mean x‾\overline{x} is:

Variance=1n∑i=1n(xi−x‾)2.\text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{x})^2.

Here, the mean x‾\overline{x} is 55. Applying this to our observations:

(a−55)2+(b−55)2+(68−55)2+(44−55)2+(48−55)2+(60−55)26=194.\frac{(a - 55)^2 + (b - 55)^2 + (68 - 55)^2 + (44 - 55)^2 + (48 - 55)^2 + (60 - 55)^2}{6} = 194.

Calculate known terms in the variance expression. Evaluate each squared term involving the known observations:

(68−55)2=132=169,(44−55)2=(−11)2=121,(48−55)2=(−7)2=49,(60−55)2=52=25.(68 - 55)^2 = 13^2 = 169, \quad (44 - 55)^2 = (-11)^2 = 121, \quad (48 - 55)^2 = (-7)^2 = 49, \quad (60 - 55)^2 = 5^2 = 25.

Substitute these values into the variance equation:

(a−55)2+(b−55)2+169+121+49+256=194.\frac{(a - 55)^2 + (b - 55)^2 + 169 + 121 + 49 + 25}{6} = 194.

Simplify:

(a−55)2+(b−55)2+3646=194.\frac{(a - 55)^2 + (b - 55)^2 + 364}{6} = 194.

Multiply both sides by 6:

(a−55)2+(b−55)2+364=1164.(a - 55)^2 + (b - 55)^2 + 364 = 1164.

Subtract 364 from both sides:

(a−55)2+(b−55)2=800(Equation 2).(a - 55)^2 + (b - 55)^2 = 800 \quad \text{(Equation 2)}.

Solve the system of equations. We have the following two equations: 1. a+b=110a + b = 110. 2. (a−55)2+(b−55)2=800(a - 55)^2 + (b - 55)^2 = 800.

From Equation 1, express aa in terms of bb:

a=110−b.a = 110 - b.

Substitute a=110−ba = 110 - b into Equation 2:

(110−b−55)2+(b−55)2=800.(110 - b - 55)^2 + (b - 55)^2 = 800.

Simplify each term:

(55−b)2+(b−55)2=800.(55 - b)^2 + (b - 55)^2 = 800.

Since (55−b)2=(b−55)2(55 - b)^2 = (b - 55)^2, we can write:

2(b−55)2=800.2(b - 55)^2 = 800.

(b−55)2=400.(b - 55)^2 = 400.

Taking the square root of both sides:

b−55=±20.b - 55 = \pm 20.

This gives: 1. b=75b = 75 (if b−55=20b - 55 = 20), 2. b=35b = 35 (if b−55=−20b - 55 = -20).

Since a>ba > b, we choose b=35b = 35. Substitute b=35b = 35 into Equation 1:

a+35=110.a + 35 = 110. a=75.a = 75.

Calculate a+3ba + 3b

a+3b=75+3⋅35=75+105=180.a + 3b = 75 + 3 \cdot 35 = 75 + 105 = 180.

Thus, the answer is: 180.