A frequency distribution table displays the frequency of each data element in the dataset.
The dataset is given as:
[tex]\left[\begin{array}{cc}Years & Frequency &Under\ 4& 21.5 & 5-14 & 39.9 &15-19 & 20.3 & 20-24 & 22.3 & 25-34 & 48.4 &35-44 & 37.8 & 45-64 & 75.7 & 65 - over & 54.3\end{array}\right][/tex]
(a) The frequency distribution table
The frequency distribution table should include, the years, the frequency, and the class midpoint (x)
The class midpoint is the average of each class
For instance;
The midpoint of under 4 is:
[tex]x = \frac{0 + 4}{2} = 2[/tex]
The midpoint of 5-14 is:
[tex]x = \frac{5 + 14}{2} = 9.5[/tex]
And so on.....
So, we have:
[tex]\left[\begin{array}{ccc}Years & Frequency & x & Under\ 4& 21.5 & 2 & 5-14 & 39.9&9.5 &15-19 & 20.3&17 & 20-24 & 22.3 & 22 & 25-34 & 48.4& 29.5 &35-44 & 37.8& 39.5 & 45-64 & 75.7 & 54.5 & 65 - over & 54.3 & 70\end{array}\right][/tex]
(b) Sample mean and Sample standard deviation
The sample mean is calculated using:
[tex]\bar x = \frac{\sum fx}{\sum f}[/tex]
So, we have:
[tex]\bar x = \frac{21.5 \times 2 + 39.9 \times 9.5 + 20.3 \times 17 + 22.3 \times 22 + 48.4 \times 29.5 + 37.8 \times 39.5 + 75.7 \times 54.5 + 54.3 \times 70}{21.5 + 39.9 + 20.3 + 22.3 + 48.4 + 37.8 + 75.7 + 54.3}[/tex]
[tex]\bar x = \frac{12105.3}{320.2}[/tex]
[tex]\bar x = 37.81[/tex]
The sample standard deviation is calculated using:
[tex]\sigma_x = \sqrt\frac{\sum f(x - \bar x)^2}{\sum f - 1}[/tex]
So, we have:
[tex]\sigma_x = \sqrt{\frac{21.5 \times (2 - 37.81)^2 + 39.9 \times (9.5 - 37.81)^2+..................+ 75.7 \times (54.5 - 37.81)^2+ 54.3 \times (70- 37.81)^2}{21.5 + 39.9 + 20.3 + 22.3 + 48.4 + 37.8 + 75.7 + 54.3 - 1}}[/tex]
[tex]\sigma_x = \sqrt\frac{154716.23522}{319.2}[/tex]
[tex]\sigma_x = \sqrt{484.699985025}[/tex]
[tex]\sigma_x = 22.02[/tex]
Hence, the sample mean and the sample standard deviation are 37.81 and 22.02, respectively.
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