`{:("Class Interval",1-5,6-10,11-15,16-20),("Frequency",3,7,6,5):}`
Consider the following statement is respect of the above frequency distribution.
I. The median is contained in the modal class.
II. The distribution is bell-shaped.
Which of the above statement is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Explanation

To solve the problem, we need to analyze the given frequency distribution and evaluate the two statements regarding the median and the shape of the distribution.

### Step 1: Define the Class Intervals and Frequencies
The class intervals and their corresponding frequencies are as follows:
- Class Interval: 1-5, 6-10, 11-15, 16-20
- Frequency: 3, 7, 6, 5

### Step 2: Adjust the Class Intervals
To make the class intervals continuous, we can adjust them by adding and subtracting 0.5:
- Adjusted Class Intervals:
- 0.5 to 5.5
- 5.5 to 10.5
- 10.5 to 15.5
- 15.5 to 20.5

### Step 3: Calculate the Cumulative Frequency
Now, we calculate the cumulative frequency:
- For the first interval (0.5 to 5.5): 3
- For the second interval (5.5 to 10.5): 3 + 7 = 10
- For the third interval (10.5 to 15.5): 10 + 6 = 16
- For the fourth interval (15.5 to 20.5): 16 + 5 = 21

So, the cumulative frequencies are:
- 3, 10, 16, 21

### Step 4: Calculate n and n/2
The total number of observations (n) is the sum of the frequencies:
- n = 3 + 7 + 6 + 5 = 21

Now, calculate n/2:
- n/2 = 21 / 2 = 10.5

### Step 5: Identify the Median Class
To find the median class, we look for the cumulative frequency that is just greater than n/2 (10.5):
- The cumulative frequency just greater than 10.5 is 16, which corresponds to the class interval 10.5 to 15.5.

### Step 6: Calculate the Median
Using the median formula:
\[ \text{Median} = L + \left(\frac{n/2 - CF}{f}\right) \times h \]
where:
- \(L\) = lower limit of the median class = 10.5
- \(CF\) = cumulative frequency of the class before the median class = 10
- \(f\) = frequency of the median class = 6
- \(h\) = width of the class interval = 5 (from 10.5 to 15.5)

Substituting the values:
\[ \text{Median} = 10.5 + \left(\frac{10.5 - 10}{6}\right) \times 5 \]
\[ = 10.5 + \left(\frac{0.5}{6}\right) \times 5 \]
\[ = 10.5 + \frac{2.5}{6} \]
\[ = 10.5 + 0.4167 \approx 10.917 \]

### Step 7: Evaluate the Statements
1. **Statement I**: The median is contained in the modal class.
- The modal class is the class with the highest frequency, which is 6-10 (frequency 7). The median (10.917) is not in this class. Therefore, this statement is **false**.

2. **Statement II**: The distribution is bell-shaped.
- To determine if the distribution is bell-shaped, we look at the cumulative frequency. The cumulative frequencies are increasing (3, 10, 16, 21), which does not form a bell shape. Therefore, this statement is also **false**.

### Conclusion
Both statements are incorrect.

### Final Answer
Neither statement I nor II is correct.

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