Answer :
Certainly! Let's work through the steps to complete both the table and the histogram.
### Part (a) Completing the Table
From the provided data, we have the waiting times and their corresponding frequencies:
1. Interval [tex]\( 0 < t < 30 \)[/tex]: The frequency is 15.
2. Interval [tex]\( 30 < t < 40 \)[/tex]: The frequency is 35.
3. Interval [tex]\( 40 < t < 60 \)[/tex]: The frequency is 50.
4. Interval [tex]\( 60 \leq t < 80 \)[/tex]: The frequency is 20.
5. Interval [tex]\( 80 < t < 100 \)[/tex]: The frequency is 10.
These values are already listed, so we can confirm the table is complete as provided:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time} \; ( t \; \text{minutes}) & \text{Frequency} \\ \hline 0 < t < 30 & 15 \\ \hline 30 < t < 40 & 35 \\ \hline 40 < t < 60 & 50 \\ \hline 60 \leqslant t < 80 & 20 \\ \hline 80 < t < 100 & 10 \\ \hline \end{array} \][/tex]
### Part (b) Completing the Histogram
To complete the histogram, we need to plot the frequencies for each time interval on a graph. Here’s a step-by-step guide:
1. Label the X-axis and Y-axis:
- X-axis: Represents the time intervals in minutes.
- Y-axis: Represents the frequency of patients.
2. Mark the intervals on the X-axis:
- Divide the X-axis into the given time intervals: [tex]\( 0 < t < 30 \)[/tex], [tex]\( 30 < t < 40 \)[/tex], [tex]\( 40 < t < 60 \)[/tex], [tex]\( 60 \leq t < 80 \)[/tex], and [tex]\( 80 < t < 100 \)[/tex].
3. Plot the frequencies as bars for each interval:
- For [tex]\( 0 < t < 30 \)[/tex], draw a bar up to the frequency of 15.
- For [tex]\( 30 < t < 40 \)[/tex], draw a bar up to the frequency of 35.
- For [tex]\( 40 < t < 60 \)[/tex], draw a bar up to the frequency of 50.
- For [tex]\( 60 \leq t < 80 \)[/tex], draw a bar up to the frequency of 20.
- For [tex]\( 80 < t < 100 \)[/tex], draw a bar up to the frequency of 10.
4. Ensure the heights of the bars correspond to the frequencies:
- Bar height for [tex]\( 0 < t < 30 \)[/tex] should be 15.
- Bar height for [tex]\( 30 < t < 40 \)[/tex] should be 35.
- Bar height for [tex]\( 40 < t < 60 \)[/tex] should be 50.
- Bar height for [tex]\( 60 \leq t < 80 \)[/tex] should be 20.
- Bar height for [tex]\( 80 < t < 100 \)[/tex] should be 10.
Ensure to use an appropriate scale on the Y-axis so that all frequencies fit well within the graph.
### Summary
- The table of waiting times and their frequencies is already complete.
- To complete the histogram, plot the intervals on the X-axis and draw bars with heights corresponding to the frequencies on the Y-axis.
This completes parts (a) and (b) effectively.
### Part (a) Completing the Table
From the provided data, we have the waiting times and their corresponding frequencies:
1. Interval [tex]\( 0 < t < 30 \)[/tex]: The frequency is 15.
2. Interval [tex]\( 30 < t < 40 \)[/tex]: The frequency is 35.
3. Interval [tex]\( 40 < t < 60 \)[/tex]: The frequency is 50.
4. Interval [tex]\( 60 \leq t < 80 \)[/tex]: The frequency is 20.
5. Interval [tex]\( 80 < t < 100 \)[/tex]: The frequency is 10.
These values are already listed, so we can confirm the table is complete as provided:
[tex]\[ \begin{array}{|c|c|} \hline \text{Time} \; ( t \; \text{minutes}) & \text{Frequency} \\ \hline 0 < t < 30 & 15 \\ \hline 30 < t < 40 & 35 \\ \hline 40 < t < 60 & 50 \\ \hline 60 \leqslant t < 80 & 20 \\ \hline 80 < t < 100 & 10 \\ \hline \end{array} \][/tex]
### Part (b) Completing the Histogram
To complete the histogram, we need to plot the frequencies for each time interval on a graph. Here’s a step-by-step guide:
1. Label the X-axis and Y-axis:
- X-axis: Represents the time intervals in minutes.
- Y-axis: Represents the frequency of patients.
2. Mark the intervals on the X-axis:
- Divide the X-axis into the given time intervals: [tex]\( 0 < t < 30 \)[/tex], [tex]\( 30 < t < 40 \)[/tex], [tex]\( 40 < t < 60 \)[/tex], [tex]\( 60 \leq t < 80 \)[/tex], and [tex]\( 80 < t < 100 \)[/tex].
3. Plot the frequencies as bars for each interval:
- For [tex]\( 0 < t < 30 \)[/tex], draw a bar up to the frequency of 15.
- For [tex]\( 30 < t < 40 \)[/tex], draw a bar up to the frequency of 35.
- For [tex]\( 40 < t < 60 \)[/tex], draw a bar up to the frequency of 50.
- For [tex]\( 60 \leq t < 80 \)[/tex], draw a bar up to the frequency of 20.
- For [tex]\( 80 < t < 100 \)[/tex], draw a bar up to the frequency of 10.
4. Ensure the heights of the bars correspond to the frequencies:
- Bar height for [tex]\( 0 < t < 30 \)[/tex] should be 15.
- Bar height for [tex]\( 30 < t < 40 \)[/tex] should be 35.
- Bar height for [tex]\( 40 < t < 60 \)[/tex] should be 50.
- Bar height for [tex]\( 60 \leq t < 80 \)[/tex] should be 20.
- Bar height for [tex]\( 80 < t < 100 \)[/tex] should be 10.
Ensure to use an appropriate scale on the Y-axis so that all frequencies fit well within the graph.
### Summary
- The table of waiting times and their frequencies is already complete.
- To complete the histogram, plot the intervals on the X-axis and draw bars with heights corresponding to the frequencies on the Y-axis.
This completes parts (a) and (b) effectively.
