Answer :
Sure, let's go through the steps one by one.
### Part (A)
We are given:
[tex]\[ \Sigma (fm) = 5k + 25 \][/tex]
[tex]\[ \Sigma f = k + 5 \][/tex]
The mean (μ) in a dataset is given by the formula:
[tex]\[ \mu = \frac{\Sigma (fm)}{\Sigma f} \][/tex]
Substituting the given values into the formula:
[tex]\[ \mu = \frac{5k + 25}{k + 5} \][/tex]
So, the mean of the data is:
[tex]\[ \mu = \frac{5k + 25}{k + 5} \][/tex]
### Part (B)
In a grouped data, we are given the mean ([tex]\(\overline{x}\)[/tex]):
[tex]\[ \overline{x} = 35.5 \][/tex]
and the total number of observations (N):
[tex]\[ N = 14 \][/tex]
The value of [tex]\(\Sigma f\)[/tex] (which typically represents the total frequency) can be found by multiplying the mean by the total number of observations:
[tex]\[ \Sigma f = \overline{x} \times N \][/tex]
Substituting the given values:
[tex]\[ \Sigma f = 35.5 \times 14 \][/tex]
[tex]\[ \Sigma f = 497.0 \][/tex]
So, the value of [tex]\(\Sigma f\)[/tex] is 497.0.
### Part (C)
In a grouped data, [tex]\(\Sigma (fm)\)[/tex] represents the sum of the product of the frequencies and the midpoints of the class intervals. Essentially, it is the weighted sum of the midpoints with the frequencies as weights, which is used in the calculation of the mean. This term helps in determining the average value of the dataset when the data is grouped into intervals, often necessary in statistical analysis.
So summarizing the answers:
(A) The mean of the data is:
[tex]\[ \mu = \frac{5k + 25}{k + 5} \][/tex]
(B) The value of [tex]\(\Sigma f\)[/tex] is:
[tex]\[ \Sigma f = 497.0 \][/tex]
(C) [tex]\(\Sigma (fm)\)[/tex] represents the sum of the product of the frequencies and the midpoints of the class intervals in a grouped data.
### Part (A)
We are given:
[tex]\[ \Sigma (fm) = 5k + 25 \][/tex]
[tex]\[ \Sigma f = k + 5 \][/tex]
The mean (μ) in a dataset is given by the formula:
[tex]\[ \mu = \frac{\Sigma (fm)}{\Sigma f} \][/tex]
Substituting the given values into the formula:
[tex]\[ \mu = \frac{5k + 25}{k + 5} \][/tex]
So, the mean of the data is:
[tex]\[ \mu = \frac{5k + 25}{k + 5} \][/tex]
### Part (B)
In a grouped data, we are given the mean ([tex]\(\overline{x}\)[/tex]):
[tex]\[ \overline{x} = 35.5 \][/tex]
and the total number of observations (N):
[tex]\[ N = 14 \][/tex]
The value of [tex]\(\Sigma f\)[/tex] (which typically represents the total frequency) can be found by multiplying the mean by the total number of observations:
[tex]\[ \Sigma f = \overline{x} \times N \][/tex]
Substituting the given values:
[tex]\[ \Sigma f = 35.5 \times 14 \][/tex]
[tex]\[ \Sigma f = 497.0 \][/tex]
So, the value of [tex]\(\Sigma f\)[/tex] is 497.0.
### Part (C)
In a grouped data, [tex]\(\Sigma (fm)\)[/tex] represents the sum of the product of the frequencies and the midpoints of the class intervals. Essentially, it is the weighted sum of the midpoints with the frequencies as weights, which is used in the calculation of the mean. This term helps in determining the average value of the dataset when the data is grouped into intervals, often necessary in statistical analysis.
So summarizing the answers:
(A) The mean of the data is:
[tex]\[ \mu = \frac{5k + 25}{k + 5} \][/tex]
(B) The value of [tex]\(\Sigma f\)[/tex] is:
[tex]\[ \Sigma f = 497.0 \][/tex]
(C) [tex]\(\Sigma (fm)\)[/tex] represents the sum of the product of the frequencies and the midpoints of the class intervals in a grouped data.
