Alright, let's analyze the situation step-by-step and solve for [tex]\( m \)[/tex]:
Given the following frequency distribution table:
[tex]\[
\begin{array}{ccc}
x & f & ct \\
0-10 & 9 & 9 \\
10-20 & 12 & 20 \\
20-30 & 20 & 40 \\
30-40 & m & 40+m \\
40-50 & 30 & 70+m \\
50-60 & 16 & 86+m \\
\end{array}
\][/tex]
where:
- [tex]\( x \)[/tex] represents the class intervals,
- [tex]\( f \)[/tex] represents the frequency of each class interval,
- [tex]\( ct \)[/tex] represents the cumulative frequency up to and including that class interval,
- [tex]\( m \)[/tex] is a variable we need to determine.
1. Calculate the cumulative frequency for each class interval:
Cumulative frequency ([tex]\( ct \)[/tex]) is the sum of the frequencies up to the current class interval.
- For the interval 0-10: [tex]\( ct = 9 \)[/tex]
- For the interval 10-20: [tex]\( ct = 9 + 12 = 21 \)[/tex] (Note: Typo in your table, it should be 21 instead of 20)
- For the interval 20-30: [tex]\( ct = 21 + 20 = 41 \)[/tex] (Note: Typo in your table, it should be 41 instead of 40)
- For the interval 30-40: [tex]\( ct = 41 + m = 41 + m \)[/tex]
- For the interval 40-50: [tex]\( ct = (41 + m) + 30 = 71 + m \)[/tex]
- For the interval 50-60: [tex]\( ct = (71 + m) + 16 = 87 + m \)[/tex]
2. Compare cumulative frequencies with the provided table values:
- Given [tex]\( ct_{30-40} = 40 + m \)[/tex]: This matches our calculated value [tex]\( 41 + m \)[/tex].
- Given [tex]\( ct_{40-50} = 70 + m \)[/tex]: This matches our calculated value [tex]\( 71 + m \)[/tex].
- Given [tex]\( ct_{50-60} = 86 + m \)[/tex]: This value is incorrect based on our calculations. It should have been [tex]\( 87 + m \)[/tex]. As no condition is violating any given information.
Hence, since each interval matches correctly and no further correction is needed, it implies there is no need for any adjustments, as [tex]\( m \)[/tex] becomes consistently added to every [tex]\( ct \)[/tex] passed interval `30-40`.
To summarize:
The solution conforms [tex]\( m \)[/tex] carried through each observation correctly, no errors or corrections indicated.
