Problem 68 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Suppose a population \(A\) has 100 observations 101,102 , ............., 200 and another population B has 100 obsevrations \(151,152, \ldots \ldots \ldots \ldots . .250\). If \(V_{A}\) and \(V_{\mathrm{B}}\) represent the variances of the two populations, respectively then \(\frac{V_{A}}{V_{B}}\) is [2006] (a) 1 (b) \(\frac{9}{4}\) (c) \(\frac{4}{9}\) (d) \(\frac{2}{3}\)

Step-by-Step Solution

Answer

(a) 1

Step 1: Define the Range of Each Population

Population A has observations from 101 to 200, inclusive, forming an arithmetic sequence with the first term (\(a_1\)) as 101 and the last term (\(a_n\)) as 200. Population B has observations from 151 to 250, with a similar arithmetic sequence beginning at 151 and ending at 250.

Step 2: Calculate the Mean of Population A

To find the mean of Population A, use the formula for the mean of an arithmetic sequence: \(\text{mean} = \frac{\text{first term} + \text{last term}}{2}\). Thus, the mean for A is \(\frac{101 + 200}{2} = 150.5\).

Step 3: Calculate the Mean of Population B

Use the same mean formula for Population B: \(\text{mean} = \frac{151 + 250}{2} = 200.5\).

Step 4: Calculate the Variance of Population A

Variance is calculated as \(V = \frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2\). For A, determine the variance using the fact that arithmetic sequences have equal intervals: \(V_A = \frac{((n^2 - 1))}{12}\). Since \(n = 100\), calculate: \(V_A = \frac{9999}{12} = 833.25\).

Step 5: Calculate the Variance of Population B

Apply the same formula used in Step 4 for Population B: \(V_B = \frac{9999}{12} = 833.25\). The variance for B is the same since it also consists of 100 observations evenly spaced by 1 unit.

Step 6: Calculate the Ratio of Variances

The ratio \(\frac{V_A}{V_B}\) is given by dividing the variances: \(\frac{833.25}{833.25} = 1\).

Step 7: Conclusion

Since \(\frac{V_A}{V_B} = 1\), the ratio of variances for the two populations is equal, indicating that the two variances are identical.

Key Concepts

Variance of Population

Variance indicates how much the data points in a population differ from the mean. In simpler terms, it measures the spread of the data. For any given set of values, like a whole population, the population variance can be calculated by

• First finding the difference between each data point and the mean.
• Then squaring these differences.
• Summing them up, and dividing by the number of data points.

This formula can be expressed as:\[V = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\]where \(x_i\) are the individual data points and \(\bar{x}\) is the mean. The higher the variance, the more spread out the population data is. In our exercise, both populations have variances calculated using their arithmetic sequences, which makes them identical in their spread.

Arithmetic Sequence

An arithmetic sequence is a series of numbers with a constant difference between consecutive terms. For instance, the populations in the exercise, 101, 102, ..., 200 and 151, 152, ..., 250, are arithmetic sequences:

• Population A starts at 101 and ends at 200
• Population B starts at 151 and ends at 250

Each sequence has 100 terms, produced by adding 1 to each previous term. Arithmetic sequences are straightforward to manage mathematically, especially when dealing with population data, because their regular increments simplify calculations like mean and variance.

Mean Calculation

The mean, or average, of a set of numbers is one of the most fundamental concepts in statistics. To calculate the mean of an arithmetic sequence:

• Add the first and the last term in the sequence.
• Divide the result by 2.

For Population A: \(\text{mean} = \frac{101 + 200}{2} = 150.5\).For Population B:\(\text{mean} = \frac{151 + 250}{2} = 200.5\).The mean serves as a central value, around which other data points are spread, determining an understanding of the dataset's overall tendency.

Population Variance Ratio

The comparison of variances for different populations is expressed as a ratio, known as the population variance ratio. This ratio can reveal how two datasets compare in terms of variability.

• If the ratio equals 1, like in this exercise, it indicates that the populations have identical variances, or spreads.
• Any difference in this ratio would suggest one population is more spread out, or variable, than the other.

For our exercise:\(\frac{V_A}{V_B} = 1\)This indicates that both populations have the same degree of variance, as both result from evenly spaced data points in an arithmetic sequence.