Question

(a)
plot a histogram from the data shown in
the table.

Explanation:

Step1: Calculate class widths

For \(0 < h\leq20\), width \(=20 - 0=20\).
For \(20 < h\leq40\), width \(=40 - 20 = 20\).
For \(40 < h\leq80\), width \(=80 - 40=40\).
For \(80 < h\leq90\), width \(=90 - 80 = 10\).

Step2: Calculate frequency density for \(40 < h\leq80\)

Use the formula \(frequency\ density=\frac{frequency}{class\ width}\).
So, \(frequency\ density=\frac{44}{40}=1.1\).

Step3: Calculate frequency density for \(80 < h\leq90\)

Using the formula \(frequency\ density=\frac{frequency}{class\ width}\).
So, \(frequency\ density=\frac{14}{10}=1.4\).

Step4: Plot the histogram

• On the x - axis, mark the class intervals (\(0 - 20\), \(20 - 40\), \(40 - 80\), \(80 - 90\)).
• On the y - axis, mark the frequency density values.
• For the class \(0 < h\leq20\), draw a bar with height \(1.2\) (since frequency density is \(1.2\)) over the interval \(0 - 20\).
• For the class \(20 < h\leq40\), draw a bar with height \(1.6\) over the interval \(20 - 40\).
• For the class \(40 < h\leq80\), draw a bar with height \(1.1\) over the interval \(40 - 80\).
• For the class \(80 < h\leq90\), draw a bar with height \(1.4\) over the interval \(80 - 90\).

Answer:

The completed table with frequency densities:

Height (\(cm\))	Frequency	Frequency Density
\(20 < h\leq40\)	\(32\)	\(1.6\)
\(40 < h\leq80\)	\(44\)	\(1.1\)
\(80 < h\leq90\)	\(14\)	\(1.4\)

And the histogram is plotted as described in the steps above with bars of appropriate heights over their respective class intervals.