Let's address the problem step by step:
### a) Constructing the Frequency Distribution
We are given the ticket sales in million dollars of the 15 highest grossing movies of all time. These ticket sales are:
1744, 1601, 1303, 1200, 1200, 1106, 1155, 1151, 983, 978, 939, 921, 918, 914, 900
We need to start with a first class of 877-906 and create an appropriate frequency distribution.
Here is the frequency distribution table:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Sales (million \$)} & \text{Number of Movies} \\
\hline \$877-906 & 1 \\
\hline \$907-936 & 3 \\
\hline \$937-966 & 1 \\
\hline \$967-996 & 2 \\
\hline \$997-1026 & 0 \\
\hline \$1027-1056 & 0 \\
\hline \$1057-1086 & 0 \\
\hline \$1087-1116 & 1 \\
\hline \$1117-1146 & 0 \\
\hline \$1147-1176 & 2 \\
\hline \$1177-1206 & 2 \\
\hline \$1207-1236 & 0 \\
\hline \$1237-1266 & 0 \\
\hline \$1267-1296 & 0 \\
\hline \$1297-1326 & 1 \\
\hline \$1327-1356 & 0 \\
\hline \$1357-1386 & 0 \\
\hline \$1387-1416 & 0 \\
\hline \$1417-1446 & 0 \\
\hline \$1447-1476 & 0 \\
\hline \$1477-1506 & 0 \\
\hline \$1507-1536 & 0 \\
\hline \$1537-1566 & 0 \\
\hline \$1567-1596 & 0 \\
\hline \$1597-1626 & 1 \\
\hline \$1627-1656 & 0 \\
\hline \$1657-1686 & 0 \\
\hline \$1687-1716 & 0 \\
\hline \$1717-1746 & 1 \\
\hline
\end{array}
\][/tex]
### b) Constructing the Histogram
A histogram is a type of bar chart that is used to represent the frequency distribution of a dataset. On the x-axis, we list the class intervals. On the y-axis, we list the frequency of each class. Here's how we can construct it based on our table:
1. Create a label for each class interval on the x-axis: \[tex]$877-906, \$[/tex]907-936, \[tex]$937-966, etc.
2. Set the y-axis to represent frequencies (starting from 0 to the maximum frequency value, which is 3 in this case).
3. Draw bars for each class interval with heights corresponding to their frequency.
### c) Constructing the Frequency Polygon
A frequency polygon is a graphical representation of the frequencies where points are plotted for each class interval at the midpoints and then connected with straight lines.
Steps to construct the frequency polygon:
1. Calculate the midpoint for each class interval. For the interval \$[/tex]877-906, the midpoint would be [tex]\(\frac{877 + 906}{2} = 891.5\)[/tex].
2. Plot these midpoints on the x-axis.
3. Plot the frequencies on the y-axis.
4. Connect these points with straight lines.
5. Optionally, you can start and end the graph on the x-axis baseline by connecting to the midpoints just before the first class and just after the last class (both with a frequency of 0).
Remember, the visualization part would typically be something you'd construct in a graphing tool or software. But now you should have a clear idea of how both the histogram and frequency polygon should look.
