Answer :
To solve the given problem, we need to analyze the distribution of marks among 500 students. We have to find two things:
1. The marks interval where only 20% of the students failed the examination.
2. The minimum marks for the top 25% of the students.
Here is the step-by-step solution:
### Step 1: Understanding the Data
We have the following distribution of marks:
- Marks range: [tex]\([0-20]\)[/tex], Number of students: [tex]\(50\)[/tex]
- Marks range: [tex]\([20-40]\)[/tex], Number of students: [tex]\(100\)[/tex]
- Marks range: [tex]\([40-50]\)[/tex], Number of students: [tex]\(150\)[/tex]
- Marks range: [tex]\([50-60]\)[/tex], Number of students: [tex]\(90\)[/tex]
- Marks range: [tex]\([60-80]\)[/tex], Number of students: [tex]\(60\)[/tex]
- Marks range: [tex]\([80-100]\)[/tex], Number of students: [tex]\(50\)[/tex]
### Step 2: Calculate the Cumulative Frequency
To find out some percentage of students, we first need to calculate the cumulative frequency of the students:
- Cumulative frequency for [tex]\(0-20\)[/tex]: [tex]\(50\)[/tex]
- Cumulative frequency for [tex]\(0-40\)[/tex]: [tex]\(50 + 100 = 150\)[/tex]
- Cumulative frequency for [tex]\(0-50\)[/tex]: [tex]\(150 + 150 = 300\)[/tex]
- Cumulative frequency for [tex]\(0-60\)[/tex]: [tex]\(300 + 90 = 390\)[/tex]
- Cumulative frequency for [tex]\(0-80\)[/tex]: [tex]\(390 + 60 = 450\)[/tex]
- Cumulative frequency for [tex]\(0-100\)[/tex]: [tex]\(450 + 50 = 500\)[/tex]
Thus, the cumulative frequencies are [tex]\([50, 150, 300, 390, 450, 500]\)[/tex].
### Step 3: Finding the 20% of Students that Failed
20% of the students equates to:
[tex]\[ 20\% \text{ of } 500 \text{ students} = 0.2 \times 500 = 100 \text{ students} \][/tex]
Now we need to find the marks range where the cumulative frequency reaches 100 students:
- The first cumulative frequency range (0-20) includes 50 students.
- The next cumulative frequency range (0-40) includes another 100 students (cumulative frequency is 150).
Thus, the 100th student falls within the marks range [tex]\(20-40\)[/tex]. Therefore, only 20% of the students have failed if they scored below the lower benchmark of the next interval, hence [tex]\(20\)[/tex] marks.
### Step 4: Finding the Minimum Marks for the Top 25% Students
25% of the students equates to:
[tex]\[ 25\% \text{ of } 500 \text{ students} = 0.25 \times 500 = 125 \text{ students} \][/tex]
This means the top 25% starts from the last [tex]\(125\)[/tex] students moving upward in the cumulative frequency:
[tex]\[ 500 - 125 = 375\text{th student onwards} \][/tex]
We need to find the cumulative frequency where the 375th student falls:
- The cumulative frequency just before 375 is 300 (for the marks range 40-50).
- The next cumulative frequency is 390, which includes the 375th student (marks range 50-60).
Thus, the minimum marks for the top 25% students fall within the next interval leading from [tex]\(50\)[/tex] marks onward.
### Summary:
1. Only 20% of the students had failed if they scored below [tex]\(20\)[/tex] marks.
2. The minimum marks required to be in the top 25% of the students is [tex]\(50\)[/tex] marks.
1. The marks interval where only 20% of the students failed the examination.
2. The minimum marks for the top 25% of the students.
Here is the step-by-step solution:
### Step 1: Understanding the Data
We have the following distribution of marks:
- Marks range: [tex]\([0-20]\)[/tex], Number of students: [tex]\(50\)[/tex]
- Marks range: [tex]\([20-40]\)[/tex], Number of students: [tex]\(100\)[/tex]
- Marks range: [tex]\([40-50]\)[/tex], Number of students: [tex]\(150\)[/tex]
- Marks range: [tex]\([50-60]\)[/tex], Number of students: [tex]\(90\)[/tex]
- Marks range: [tex]\([60-80]\)[/tex], Number of students: [tex]\(60\)[/tex]
- Marks range: [tex]\([80-100]\)[/tex], Number of students: [tex]\(50\)[/tex]
### Step 2: Calculate the Cumulative Frequency
To find out some percentage of students, we first need to calculate the cumulative frequency of the students:
- Cumulative frequency for [tex]\(0-20\)[/tex]: [tex]\(50\)[/tex]
- Cumulative frequency for [tex]\(0-40\)[/tex]: [tex]\(50 + 100 = 150\)[/tex]
- Cumulative frequency for [tex]\(0-50\)[/tex]: [tex]\(150 + 150 = 300\)[/tex]
- Cumulative frequency for [tex]\(0-60\)[/tex]: [tex]\(300 + 90 = 390\)[/tex]
- Cumulative frequency for [tex]\(0-80\)[/tex]: [tex]\(390 + 60 = 450\)[/tex]
- Cumulative frequency for [tex]\(0-100\)[/tex]: [tex]\(450 + 50 = 500\)[/tex]
Thus, the cumulative frequencies are [tex]\([50, 150, 300, 390, 450, 500]\)[/tex].
### Step 3: Finding the 20% of Students that Failed
20% of the students equates to:
[tex]\[ 20\% \text{ of } 500 \text{ students} = 0.2 \times 500 = 100 \text{ students} \][/tex]
Now we need to find the marks range where the cumulative frequency reaches 100 students:
- The first cumulative frequency range (0-20) includes 50 students.
- The next cumulative frequency range (0-40) includes another 100 students (cumulative frequency is 150).
Thus, the 100th student falls within the marks range [tex]\(20-40\)[/tex]. Therefore, only 20% of the students have failed if they scored below the lower benchmark of the next interval, hence [tex]\(20\)[/tex] marks.
### Step 4: Finding the Minimum Marks for the Top 25% Students
25% of the students equates to:
[tex]\[ 25\% \text{ of } 500 \text{ students} = 0.25 \times 500 = 125 \text{ students} \][/tex]
This means the top 25% starts from the last [tex]\(125\)[/tex] students moving upward in the cumulative frequency:
[tex]\[ 500 - 125 = 375\text{th student onwards} \][/tex]
We need to find the cumulative frequency where the 375th student falls:
- The cumulative frequency just before 375 is 300 (for the marks range 40-50).
- The next cumulative frequency is 390, which includes the 375th student (marks range 50-60).
Thus, the minimum marks for the top 25% students fall within the next interval leading from [tex]\(50\)[/tex] marks onward.
### Summary:
1. Only 20% of the students had failed if they scored below [tex]\(20\)[/tex] marks.
2. The minimum marks required to be in the top 25% of the students is [tex]\(50\)[/tex] marks.
