Problem 41 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The outcome of each of 30 items was observed; 10 items gave an outcome \(\frac{1}{2}-\mathrm{d}\) each, 10 items gave outcome \(\frac{1}{2}\) each and the remaining 10 items gave outcome \(\frac{1}{2}+\mathrm{d}\) each. If the variance of this outcome data is \(\frac{4}{3}\) then \(|\mathrm{d}|\) equals : [Jan. 11, 2019 (I)] (a) \(\frac{2}{3}\) (b) 2 (c) \(\frac{\sqrt{5}}{2}\) (d) \(\sqrt{2}\)

Step-by-Step Solution

Answer

\(|d| = \sqrt{2}\), corresponding to option (d).

Step 1: Understanding the Problem

We have 30 observations divided into three groups of 10, each with outcomes \(\frac{1}{2}-\mathrm{d}\), \(\frac{1}{2}\), and \(\frac{1}{2}+\mathrm{d}\) respectively. We need to calculate the variance and find the value of \(|\mathrm{d}|\) when the variance is given as \(\frac{4}{3}\).

Step 2: Calculate the Mean (M)

Calculate the mean (\(M\)) of these values. The mean is calculated as follows:\[M = \frac{10 \cdot \left(\frac{1}{2} - \mathrm{d}\right) + 10 \cdot \left(\frac{1}{2}\right) + 10 \cdot \left(\frac{1}{2} + \mathrm{d}\right)}{30}\]Simplify to get:\[M = \frac{1}{2}\]

Step 3: Determine the Formula for Variance

The variance is calculated using the formula:\[\text{Variance} = \frac{1}{n} \sum (x_i - M)^2\]where \(x_i\) represents each observation, and \(n = 30\).

Step 4: Calculate Each Term in the Variance

Now calculate each term squared and sum them:- For the first group \((\frac{1}{2} - \mathrm{d})\): \[10 \times \left( \left( \frac{1}{2} - \mathrm{d} - \frac{1}{2} \right)^2 \right) = 10d^2\]- For the second group \((\frac{1}{2})\): \[10 \times \left( \frac{1}{2} - \frac{1}{2} \right)^2 = 0\]- For the third group \((\frac{1}{2} + \mathrm{d})\): \[10 \times \left( \left( \frac{1}{2} + \mathrm{d} - \frac{1}{2} \right)^2 \right) = 10d^2\]Thus, the total is:\[20d^2\]

Step 5: Set the Variance Equation

Set the calculated variance equal to the given variance:\[\frac{20d^2}{30} = \frac{4}{3}\]

Step 6: Solve for \(d\)

Simplify and solve the equation:\[20d^2 = \frac{4}{3} \times 30\]\[20d^2 = 40\]\[d^2 = 2\]\[|d| = \sqrt{2}\]

Step 7: Evaluate the Options

Compare this result \(|d| = \sqrt{2}\) with the given options:(a) \(\frac{2}{3}\) (b) 2 (c) \(\frac{\sqrt{5}}{2}\) (d) \(\sqrt{2}\)The closest or accurate answer is \(\sqrt{2}\).

Key Concepts

Understanding the Mean

The mean is a fundamental concept in statistics that represents the average value of a set of numbers. It provides a central value where all data points tend to cluster around. In mathematical terms, the mean is the sum of all observations divided by the number of observations. In the given problem, the data set is divided into three parts with 10 items each:

• 10 items with outcome \(\frac{1}{2} - d\)
• 10 items with outcome \(\frac{1}{2}\)
• 10 items with outcome \(\frac{1}{2} + d\)

To calculate their mean, sum up all the outcomes and divide by 30, the total number of items. The calculation is shown as:\[M = \frac{10 \left(\frac{1}{2} - d\right) + 10 \left(\frac{1}{2}\right) + 10 \left(\frac{1}{2} + d\right)}{30} = \frac{1}{2}\]The mean effectively balances the lower and higher values, proving symmetrical distribution around \(\frac{1}{2}\). Understanding the mean assists in analyzing data distributions and is pivotal for subsequent statistical calculations like variance.

Grasping the Concept of Standard Deviation

Standard deviation is a statistical measurement that reflects how much variability or dispersion exists within a set of data points. It's a crucial concept that spontaneously follows the understanding of variance. The smaller the standard deviation, the closer data points are to the mean.Variance, calculated first, measures the average squared deviation from the mean. Standard deviation, on the other hand, is simply the square root of the variance, bringing the units back to the original data unit's scale. In the given exercise, the variance was given as \(\frac{4}{3}\), and if we were to calculate the standard deviation, we would take the square root of this value. Using the formula:\[\text{Standard Deviation} = \sqrt{\text{Variance}}\]Since the variance is \(\frac{4}{3}\), the standard deviation would be:\[\text{Standard Deviation} = \sqrt{\frac{4}{3}}\]Standard deviation provides vital insights into how data is spread around the mean and signifies whether the data points are generally near the mean or widely dispersed.

Mastering Mathematical Problem-Solving

Mathematical problem-solving is a skill that enables individuals to use mathematical concepts and tools to find solutions to a variety of problems. It involves a series of logical steps that lead to the solution, which can be demonstrated by the problem provided.To tackle a problem effectively, one should initially understand the problem thoroughly. In the exercise, we explored the number of outcomes and the variance. Once we've defined what is asked, breaking down the problem into smaller manageable steps is essential. Each step should be approached with simplicity and logic:

• Translate the words into mathematical equations.
• Clearly understand and apply formulas such as for mean and variance.
• Exploit algebraic manipulations to isolate the unknowns, in this case, \(|d|\).
• Compare results with available options to ensure accuracy.

By systematically applying mathematical techniques and formulas, problem-solving becomes more streamlined. Using this structured approach facilitates a clearer path to discovering correct solutions in mathematical contexts.