Nishisharma0508idn
Answered

Get personalized answers to your unique questions on IDNLearn.com. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.

2. The given distribution shows the number of wickets taken by bowlers in one-day international cricket matches:

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\begin{tabular}{c}
Number of \\
wickets
\end{tabular} &
\begin{tabular}{c}
Less \\
than 15
\end{tabular} &
\begin{tabular}{c}
Less \\
than 30
\end{tabular} &
\begin{tabular}{c}
Less \\
than 45
\end{tabular} &
\begin{tabular}{c}
Less \\
than 60
\end{tabular} &
\begin{tabular}{c}
Less \\
than 75
\end{tabular} &
\begin{tabular}{c}
Less \\
than 90
\end{tabular} &
\begin{tabular}{c}
Less \\
than 105
\end{tabular} &
\begin{tabular}{c}
Less \\
than 120
\end{tabular} \\
\hline
\begin{tabular}{c}
Number of \\
bowlers
\end{tabular} & 2 & 5 & 9 & 17 & 39 & 54 & 70 & 80 \\
\hline
\end{tabular}

Draw a 'less than type' ogive from the above data. Find the median.

[CBSE 2014]

Sagot :

GinnyAnsweridn
Certainly! Let's work through the problem step by step.

### Drawing the 'Less than type' Ogive

1. Gather the Data:
We have two sets of data: the number of wickets and the cumulative number of bowlers.

- Number of wickets: \( 15, 30, 45, 60, 75, 90, 105, 120 \)
- Number of bowlers: \( 2, 5, 9, 17, 39, 54, 70, 80 \)

2. Setting up the Plot:
- Plot the number of wickets on the x-axis.
- Plot the cumulative number of bowlers on the y-axis.

3. Plot the Points:
We'll mark the points corresponding to each pair \((number of wickets, number of bowlers)\) using `(x, y)` form:
- (15, 2)
- (30, 5)
- (45, 9)
- (60, 17)
- (75, 39)
- (90, 54)
- (105, 70)
- (120, 80)

4. Join the Points:
Use a smooth curve to connect these points, which forms our 'Less than type' ogive.

### Finding the Median

To find the median from the ogive:

1. Total Number of Bowlers (\(N\)):
The total number of bowlers is \( 80 \).

2. Median Position:
The median position is at \( \frac{N}{2} = \frac{80}{2} = 40 \).

3. Locate Median Class:
Find the class interval where the cumulative frequency just exceeds or includes 40. Here, the class interval is 60-75 wickets.

4. Details of Median Class:
Let's break down the details needed for interpolation:
- Cumulative frequency just before the median class (CF): 17
- Frequency of the median class (f): \( 39 - 17 = 22 \) (the difference between CF at 75 and CF at 60)
- Lower class boundary of the median class (\(L\)): 60
- Class width (\(h\)): \( 75 - 60 = 15 \)

5. Apply the Median Formula:
The formula for calculating the median in a grouped frequency distribution is:
[tex]\[ \text{Median} = L + \left(\frac{\frac{N}{2} - \text{CF}}{f}\right) \times h \][/tex]
Substitute the values:
[tex]\[ \text{Median} = 60 + \left(\frac{40 - 17}{22}\right) \times 15 \][/tex]
Simplify inside the parenthesis:
[tex]\[ = 60 + \left(\frac{23}{22}\right) \times 15 \][/tex]
Calculate the fraction and the multiplication:
[tex]\[ = 60 + (1.045) \times 15 \][/tex]
[tex]\[ = 60 + 15.675 \][/tex]
[tex]\[ = 75.675 \][/tex]

So, the median number of wickets is approximately 75.68.

### Summary:
- Ogive: A plot connecting (15, 2), (30, 5), (45, 9), (60, 17), (75, 39), (90, 54), (105, 70), and (120, 80) with a smooth curve.
- Median: The median number of wickets taken by the bowlers is approximately 75.68 wickets.
Your presence in our community is highly appreciated. Keep sharing your insights and solutions. Together, we can build a rich and valuable knowledge resource for everyone. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.