Let's solve this problem step by step to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given the information:
1. Understand the problem:
- The mode is given as 37.5.
- The total frequency is 169.
- We have frequencies for several class intervals, with two unknown frequencies [tex]\( X \)[/tex] and [tex]\( Y \)[/tex].
2. Identify the modal class:
- The mode is 37.5, which falls within the class interval [tex]\( 30-40 \)[/tex]. Hence [tex]\( 30-40 \)[/tex] is the modal class.
3. Given data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\text{C.I} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 & 50-60 & 60-70 & 70-80 \\
\hline
\text{F} & 5 & 18 & X & 45 & 40 & Y & 10 & 6 \\
\hline
\end{array}
\][/tex]
- Known frequencies: [tex]\( 5, 18, 45, 40, 10, 6 \)[/tex]
- Unknown frequencies: [tex]\( X, Y \)[/tex]
4. Total frequency:
The sum of all frequencies should be equal to 169.
5. Calculate the sum of known frequencies:
[tex]\[
5 + 18 + 45 + 40 + 10 + 6 = 124
\][/tex]
6. Calculate the sum of unknown frequencies [tex]\( X \)[/tex] and [tex]\( Y \)[/tex]:
[tex]\[
X + Y = 169 - 124 = 45
\][/tex]
7. Determine possible values:
Since the modal class (30-40) has the highest frequency (45), we assume [tex]\( X \)[/tex] should be considerable enough to support the mode condition which states [tex]\( X \)[/tex] should be higher than the previous class frequency (18 from the 10-20 class interval).
Considering [tex]\( X > 18 \)[/tex]:
The possible pairs [tex]\((X, Y)\)[/tex] satisfying [tex]\( X + Y = 45 \)[/tex] with [tex]\( X > 18 \)[/tex] are:
[tex]\[
(19, 26), (20, 25), (21, 24), (22, 23), (23, 22), (24, 21), (25, 20), (26, 19), (27, 18), (28, 17), (29, 16), (30, 15), (31, 14), (32, 13), (33, 12), (34, 11), (35, 10), (36, 9), (37, 8), (38, 7), (39, 6), (40, 5), (41, 4), (42, 3), (43, 2), (44, 1)
\][/tex]
8. Validation:
To validate, we check against the condition that the modal class frequency [tex]\( 45 \)[/tex] aligns correctly with the calculated values. These pairs ensure that the total sum of frequencies equals 169.
Therefore, the values of [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are not unique but any pair from the set [tex]\((19, 26), (20, 25), (21, 24), \ldots, (44, 1)\)[/tex] will satisfy the given conditions.
