To fill in the missing relative frequencies and cumulative frequencies in the given frequency table, let's follow each step methodically:
1. Calculate the Relative Frequencies:
[tex]\[
\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Number of Firms}}
\][/tex]
Given that the total number of firms is 96, we will use this number to calculate the relative frequency for each age group and round them to four decimal places.
- For Age 40-44:
[tex]\[
\text{Relative Frequency} = \frac{16}{96} = 0.1667
\][/tex]
- For Age 45-49:
[tex]\[
\text{Relative Frequency} = \frac{17}{96} = 0.1771
\][/tex]
- For Age 50-54:
[tex]\[
\text{Relative Frequency} = \frac{9}{96} = 0.0938
\][/tex]
- For Age 55-59:
[tex]\[
\text{Relative Frequency} = \frac{17}{96} = 0.1771
\][/tex]
- For Age 60-64:
[tex]\[
\text{Relative Frequency} = \frac{16}{96} = 0.1667
\][/tex]
- For Age 65-69:
[tex]\[
\text{Relative Frequency} = \frac{12}{96} = 0.1250
\][/tex]
- For Age 70-74:
[tex]\[
\text{Relative Frequency} = \frac{9}{96} = 0.0938
\][/tex]
2. Calculate the Cumulative Frequencies:
[tex]\[
\text{Cumulative Frequency} = \sum \text{Frequencies from previous groups}
\][/tex]
- For Age 40-44:
[tex]\[
\text{Cumulative Frequency} = 16
\][/tex]
- For Age 45-49:
[tex]\[
\text{Cumulative Frequency} = 16 + 17 = 33
\][/tex]
- For Age 50-54:
[tex]\[
\text{Cumulative Frequency} = 33 + 9 = 42
\][/tex]
- For Age 55-59:
[tex]\[
\text{Cumulative Frequency} = 42 + 17 = 59
\][/tex]
- For Age 60-64:
[tex]\[
\text{Cumulative Frequency} = 59 + 16 = 75
\][/tex]
- For Age 65-69:
[tex]\[
\text{Cumulative Frequency} = 75 + 12 = 87
\][/tex]
- For Age 70-74:
[tex]\[
\text{Cumulative Frequency} = 87 + 9 = 96
\][/tex]
Now, updating the table with the calculated relative and cumulative frequencies:
\begin{tabular}{|c|c|c|c|}
\hline Age & Frequency & Relative Frequency & Cumulative Frequency \\
\hline [tex]$40-$[/tex]44 & 16 & 0.1667 & 16 \\
\hline [tex]$45-$[/tex]49 & 17 & 0.1771 & 33 \\
\hline [tex]$50-$[/tex]54 & 9 & 0.0938 & 42 \\
\hline [tex]$55-$[/tex]59 & 17 & 0.1771 & 59 \\
\hline [tex]$60-$[/tex]64 & 16 & 0.1667 & 75 \\
\hline [tex]$65-$[/tex]69 & 12 & 0.1250 & 87 \\
\hline [tex]$70-$[/tex]74 & 9 & 0.0938 & 96 \\
\hline Total & 96 & & \\
\hline
\end{tabular}
Thus, the table is now complete with the relative frequencies rounded to four decimal places and the cumulative frequencies calculated accurately.
