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The marks (out of 100) scored by Nikita in a semester exam is given as 50, 60, 55, 65, 70. If Suman has scored 5 marks more than Nikita in each subject. Based on the given information, answer the subquestions. Calculate the correlation coefficient between the marks scored by Nikita and Suman.

To solve the problem given in the image, let's break it down into parts.

### Part 1: Modified Marks Calculation
Given the formula for modified marks:
\[ \text{Modified marks} = \frac{\text{Marks} \times 50}{100} \]
Let's find the modified marks for Nikita.

Nikita's original marks: 50, 60, 55, 65, 70.

Modified marks calculation:
\[ \text{Modified marks} = \frac{50 \times 50}{100} = 25 \]
\[ \text{Modified marks} = \frac{60 \times 50}{100} = 30 \]
\[ \text{Modified marks} = \frac{55 \times 50}{100} = 27.5 \]
\[ \text{Modified marks} = \frac{65 \times 50}{100} = 32.5 \]
\[ \text{Modified marks} = \frac{70 \times 50}{100} = 35 \]

So, the modified marks for Nikita are 25, 30, 27.5, 32.5, and 35.

### Part 2: Population Variance Calculation
To find the population variance of the modified marks:

1. **Find the mean of the modified marks:**
\[ \text{Mean} = \frac{25 + 30 + 27.5 + 32.5 + 35}{5} = \frac{150}{5} = 30 \]

2. **Calculate the squared differences from the mean:**
\[ (25 - 30)^2 = 25 \]
\[ (30 - 30)^2 = 0 \]
\[ (27.5 - 30)^2 = 6.25 \]
\[ (32.5 - 30)^2 = 6.25 \]
\[ (35 - 30)^2 = 25 \]

3. **Sum of squared differences:**
\[ 25 + 0 + 6.25 + 6.25 + 25 = 62.5 \]

4. **Population variance:**
\[ \text{Variance} = \frac{62.5}{5} = 12.5 \]

Thus, the population variance of the modified marks scored by Nikita is **12.5**.

### Part 3: Correlation Coefficient between Marks of Nikita and Suman
Suman's marks: Nikita's marks + 5
Nikita's marks: 50, 60, 55, 65, 70
Suman's marks: 55, 65, 60, 70, 75

The correlation coefficient, \( r \), between two sets of scores (X and Y) can be calculated using the formula:
\[ r = \frac{n(\sum{XY}) - (\sum{X})(\sum{Y})}{\sqrt{[n\sum{X^2} - (\sum{X})^2][n\sum{Y^2} - (\sum{Y})^2]}} \]

Since Suman's marks are just a linear transformation of Nikita's marks (shifted by 5), the correlation coefficient is 1.

Would you like further details or have any questions?

Here are some related questions you can ask:

1. How is the population variance different from the sample variance?
2. What is the formula for calculating the correlation coefficient in detail?
3. How do you interpret the correlation coefficient value?
4. Can we find the covariance between the marks of Nikita and Suman?
5. How would the variance change if the modification formula was different?

**Tip:** When calculating variance, always distinguish between sample variance and population variance, as their formulas differ slightly in their denominators.

Learn how to calculate the correlation coefficient between Nikita's marks and Suman's marks, given their respective scores. Understand the correlation formula and its application in statistical analysis. Suitable for college-level students and anyone interested in statistics.

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