Problem 43 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1,3 and 8 , then a ratio of other two observations is: [Jan. 10, 2019 (I)] (a) \(10: 3\) (b) \(4: 9\) (c) \(5: 8\) (d) \(6: 7\)

Step-by-Step Solution

Answer

(a) 10:3

Step 1: Define the Given Information

We have five observations, three of which are known: 1, 3, and 8. Let the two unknown observations be represented by the variables \( x \) and \( y \). The mean of the observations is given as 5, and the variance is 9.20.

Step 2: Calculate the Sum and Mean Equations

Since the mean is 5, the equation for the sum of the observations is \( \frac{1 + 3 + 8 + x + y}{5} = 5 \). Simplifying this gives:\[1 + 3 + 8 + x + y = 25\]\[x + y = 25 - 12 = 13\]

Step 3: Setup Variance Equation

The formula for variance is given by:\[\text{Variance} = \frac{1}{N} \sum{(x_i - \text{Mean})^2}\]where \( N = 5 \) (number of observations) and the mean is 5. The variance equation is:\[\frac{(1-5)^2 + (3-5)^2 + (8-5)^2 + (x-5)^2 + (y-5)^2 }{5} = 9.20\]Simplifying gives:\[\frac{16 + 4 + 9 + (x-5)^2 + (y-5)^2}{5} = 9.20\]

Key Concepts

Mean

The mean, or average, is a basic concept in statistics that helps us find the central value of a set of numbers. When we talk about the mean, we are essentially looking at the balance point, the value that represents all numbers equally. In this exercise, we calculated the mean of five observations, three of which are known. The mean was given as 5, and this tells us that if all five numbers were the same, each would be 5.

To find the mean, sum up all the numbers and then divide by the number of numbers. If we represent a set of numbers as \( x_1, x_2, ... , x_n \), the mean \( \bar{x} \) is calculated as follows:

\[ \bar{x} = \frac{x_1 + x_2 + ... + x_n}{n} \]
In our problem, we solved \( \frac{1 + 3 + 8 + x + y}{5} = 5 \) to get the sum, ensuring all numbers together average out to 5.

Knowing how to calculate and use the mean is important, as it provides a quick overview of the data set and helps to understand its general trend.

Variance

Variance measures how much the numbers in a set deviate from the mean. It gives us an idea of how spread out the numbers are. A low variance indicates the numbers are close to the mean, while a high variance shows they are more spread out.

The formula for variance \( \sigma^2 \) for a sample is:

\[ \sigma^2 = \frac{1}{N} \sum{(x_i - \bar{x})^2} \]
In this problem, the variance was given as 9.20, suggesting some level of spread because the observations deviate significantly from the mean.

We calculated the variance using known data and the unknowns \( x \) and \( y \). The variance equation becomes:

\[ \frac{(1-5)^2 + (3-5)^2 + (8-5)^2 + (x-5)^2 + (y-5)^2}{5} = 9.20 \]
Understanding variance is essential because it helps in determining the reliability of data. If the variance of a data set is too high, any predictions or conclusions drawn from it might be less trustworthy.

System of Equations

A system of equations is a set of equations with multiple variables that are solved together because the solution needs to satisfy all equations simultaneously. In the context of this exercise, we set up two crucial equations: one for the sum of the numbers based on the given mean, and another for the variance.

We first determined:

• The equation for the sum: \( x + y = 13 \)
• The variance equation with rearranged terms to find \( x \) and \( y \).

Both equations relate to each other because they involve both unknown observations \( x \) and \( y \). Solving such systems usually involves methods like substitution, elimination, or matrices, depending on complexity.

Using these equations together helped us find the missing observations and their ratio. These tools are invaluable in mathematics and real-life scenarios where multiple conditions or restrictions are at play.