Draw a histogram for the information in the table.
| Length of time $(t)$ | Frequency |
| :------------------- | :-------- |
| $0 \leqslant t<10$ | 8 |
| $10 \leqslant t<15$ | 15 |
| $15 \leqslant t<20$ | 10 |
| $20 \leqslant t<30$ | 11 |
Answer (2)
Calculate the class widths for each interval: 10 , 5 , 5 , 10 .
Calculate the frequency densities for each interval: 0.8 , 3.0 , 2.0 , 1.1 .
Draw the histogram with time intervals on the x-axis and frequency density on the y-axis.
The height of each bar represents the frequency density for that interval.
Explanation
Understanding the Problem We are given a table that shows the lengths of time some children spent watching TV last week, grouped into intervals. Our task is to draw a histogram to represent this data. A histogram is a graphical representation of data where the area of each bar is proportional to the frequency of the corresponding interval.
Calculating Class Widths and Frequency Densities To draw the histogram, we first need to calculate the class widths and the frequency densities for each interval. The class width is the range of values in each interval, and the frequency density is the frequency divided by the class width.
Calculating Class Widths The class widths are calculated as follows:
For the interval 0 ≤ t < 10 , the class width is 10 − 0 = 10 .
For the interval 10 ≤ t < 15 , the class width is 15 − 10 = 5 .
For the interval 15 ≤ t < 20 , the class width is 20 − 15 = 5 .
For the interval 20 ≤ t < 30 , the class width is 30 − 20 = 10 .
Calculating Frequency Densities The frequency densities are calculated as follows:
For the interval 0 ≤ t < 10 , the frequency density is 10 8 = 0.8 .
For the interval 10 ≤ t < 15 , the frequency density is 5 15 = 3.0 .
For the interval 15 ≤ t < 20 , the frequency density is 5 10 = 2.0 .
For the interval 20 ≤ t < 30 , the frequency density is 10 11 = 1.1 .
Drawing the Histogram Now we can draw the histogram. The x-axis represents the time intervals, and the y-axis represents the frequency density. Each bar's height corresponds to the frequency density for that interval.
The first bar spans from 0 to 10 on the x-axis and has a height of 0.8 on the y-axis.
The second bar spans from 10 to 15 on the x-axis and has a height of 3.0 on the y-axis.
The third bar spans from 15 to 20 on the x-axis and has a height of 2.0 on the y-axis.
The fourth bar spans from 20 to 30 on the x-axis and has a height of 1.1 on the y-axis.
Final Answer The histogram visually represents the distribution of time spent watching TV by the children. The height of each bar shows the frequency density, indicating how densely populated each time interval is.
Examples
Histograms are used in many real-world scenarios, such as analyzing the distribution of exam scores in a class, representing the age distribution of a population, or visualizing the frequency of different types of defects in a manufacturing process. In finance, histograms can display the distribution of stock returns, helping investors understand the risk associated with different investments. Understanding histograms helps in interpreting data and making informed decisions based on the distribution of the data.
To draw a histogram from the provided table, calculate the widths of the intervals, plot the frequencies as heights of bars, and construct the graph. The x-axis will represent the intervals and the y-axis the frequency of occurrences. This visualizes the distribution of the data effectively.
;
