The ages, in years, of 50 teachers in a school are given below : 21 37 49 27 49 42 26 33 46 40 50 29 23 24 29 31 36 22 27 38 30 26 42 39 34 23 21 32 41 46 4...
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The ages, in years, of 50 teachers in a school are given below :
21 37 49 27 49 42 26 33 46 40 50 29 23 24 29 31 36 22 27 38 30 26 42 39 34 23 21 32 41 46 46 31 33 29 28 43 47 40 34 44 26 38 34 49 45 27 25 33 39 40
(a) Form a frequency distribution table of the data using the intervals : 21 - 25, 26 - 30, 31 - 35 etc.
(b) Draw the histogram of the distribution
(c) Use your histogram to estimate the mode
(d) Calculate the mean age.
Answer Details
(a) Frequency distribution
| Age (years) | Tally | Class mark, \(x\) | Frequency, \(f\) | \(fx\) |
|---|---|---|---|---|
| 21 - 25 | ||||| || | 23 | 7 | 161 |
| 26 - 30 | ||||| ||||| | | 28 | 11 | 308 |
| 31 - 35 | ||||| |||| | 33 | 9 | 297 |
| 36 - 40 | ||||| |||| | 38 | 9 | 342 |
| 41 - 45 | ||||| | | 43 | 6 | 258 |
| 46 - 50 | ||||| ||| | 48 | 8 | 384 |
| Total | 50 | 1750 | ||
(b) Histogram
The continuous class boundaries are \(20.5\), \(25.5\), \(30.5\), \(35.5\), \(40.5\), \(45.5\) and \(50.5\). Since all class widths are 5 years, the bar heights are the frequencies.
(c) Modal age
The modal class is \(26\text{ - }30\), for which \(L=25.5\), \(f_1=11\), \(f_0=7\), \(f_2=9\), and class width \(h=5\).
\[\text{Mode}=L+\left(\frac{f_1-f_0}{(f_1-f_0)+(f_1-f_2)}\right)h\]
\[=25.5+\left(\frac{11-7}{(11-7)+(11-9)}\right)\times5=25.5+\frac{4}{6}\times5=28.8\]
Hence, the estimated mode is \(28.8\) years.
(d) Mean age
\[\bar{x}=\frac{\sum fx}{\sum f}=\frac{1750}{50}=35\]
Therefore, the mean age is 35 years.
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