Let's go through the problem step-by-step:
1. Identify the modal class:
The mode (37.5) lies in the class interval 30-40.
Therefore, the modal class is 30-40.
2. Determine the necessary elements for the mode formula:
- [tex]\( L \)[/tex]: Lower limit of the modal class = 30
- [tex]\( f_1 \)[/tex]: Frequency of the modal class = 45
- [tex]\( f_0 \)[/tex]: Frequency of the class before the modal class = [tex]\( X \)[/tex]
- [tex]\( f_2 \)[/tex]: Frequency of the class after the modal class = 40
- [tex]\( h \)[/tex]: Class width (40 - 30) = 10
3. Using the mode formula:
The mode formula is given by:
[tex]\[
\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h
\][/tex]
Substitute the known values into the formula:
[tex]\[
37.5 = 30 + \left( \frac{45 - X}{2(45) - X - 40} \right) \times 10
\][/tex]
4. Simplify and solve for [tex]\(X\)[/tex]:
[tex]\[
37.5 = 30 + \left( \frac{45 - X}{90 - X - 40} \right) \times 10
\][/tex]
[tex]\[
37.5 = 30 + \left( \frac{45 - X}{50 - X} \right) \times 10
\][/tex]
Subtract 30 from both sides:
[tex]\[
7.5 = \left( \frac{45 - X}{50 - X} \right) \times 10
\][/tex]
Divide both sides by 10:
[tex]\[
0.75 = \frac{45 - X}{50 - X}
\][/tex]
Cross-multiply to solve for [tex]\( X \)[/tex]:
[tex]\[
0.75(50 - X) = 45 - X
\][/tex]
[tex]\[
37.5 - 0.75X = 45 - X
\][/tex]
Move [tex]\( X \)[/tex] terms to one side and constant terms to the other:
[tex]\[
45 - 37.5 = 0.25X
\][/tex]
[tex]\[
7.5 = 0.25X
\][/tex]
[tex]\[
X = \frac{7.5}{0.25} = 30
\][/tex]
5. Solve for [tex]\( Y \)[/tex] using the total frequency:
The total frequency is given as 169.
The frequencies sum equation is:
[tex]\[
5 + 18 + X + 45 + 40 + Y + 10 + 6 = 169
\][/tex]
Substitute [tex]\( X = 30 \)[/tex] into the equation:
[tex]\[
5 + 18 + 30 + 45 + 40 + Y + 10 + 6 = 169
\][/tex]
Sum the known values:
[tex]\[
154 + Y = 169
\][/tex]
Solve for [tex]\( Y \)[/tex]:
[tex]\[
Y = 169 - 154 = 15
\][/tex]
6. Summarize the results:
[tex]\[
X = 30 \quad \text{and} \quad Y = 15
\][/tex]
Therefore, the values of [tex]\( X \)[/tex] and [tex]\( Y \)[/tex] are 30 and 15, respectively.
