Histograms：練習題

A histogram represents the time, \( t \) minutes, spent on homework by a group of students. The bar for the interval \( 0 < t \le 20 \) has a frequency of 60 and a height of 1.5 units. The bars for the intervals \( 20 < t \le 30 \) and \( 30 < t \le 60 \) have heights of 4.2 units and 2.4 units respectively. Calculate the total frequency of the students in the group.

The table below shows information about the masses, \(m\) grams, of 84 objects.

Mass interval: \(0 < m \le 10\), Frequency: 15
Mass interval: \(10 < m \le 25\), Frequency: 45
Mass interval: \(25 < m \le 50\), Frequency: 24

In a histogram representing this data, the height of the bar for the class interval \(10 < m \le 25\) is 12 cm. Find the height of the bar representing the class interval \(25 < m \le 50\).

The times taken for 50 students to complete a puzzle are recorded in a histogram. For the class interval \(20 < t \le 30\), the frequency is 15. Calculate the frequency density for this bar.

In a histogram, the bar for the interval \(0 < x \le 5\) has a frequency density of \(4.2\). The bar for the interval \(5 < x \le 15\) has a frequency of 36. Determine which interval represents a higher number of observations and by how many.

A histogram represents the weights of 200 items. The frequency densities for the first three intervals are: \(0 < w \le 10\) (FD = 2.4), \(10 < w \le 15\) (FD = 8.0), and \(15 < w \le 25\) (FD = 9.6). Calculate the frequency of the remaining data if the total frequency is 200.

The table shows information about the time, \(t\) minutes, taken by 122 students to solve a logic puzzle.

\(0 < t \le 10\): Frequency = 20
\(10 < t \le 25\): Frequency = \(x\)
\(25 < t \le 30\): Frequency = 30
\(30 < t \le 50\): Frequency = 24

(a) In a histogram representing this data, the height of the bar for the interval \(10 < t \le 25\) is 3.2 units. Calculate the value of \(x\).

(b) Calculate the frequency density (height of the bar) for the other three intervals.

(c) Estimate the number of students who took between 28 and 40 minutes to solve the puzzle.

The table shows the distribution of the weights, \(w\) kg, of 100 packages in a warehouse.

\(0 < w \le 10\): Frequency = 15
\(10 < w \le 25\): Frequency = 36
\(25 < w \le 40\): Frequency = 30
\(40 < w \le 60\): Frequency = 19

(a) Calculate the frequency densities for each of the four classes.
(b) In a histogram, the bar for the class \(10 < w \le 25\) has a height of 6 cm. Find the height of the bar for the class \(40 < w \le 60\).
(c) Calculate an estimate of the number of packages with a weight greater than 35 kg.