High School

Here are the hottest recorded temperatures (in [tex]^{\circ} F[/tex]) for each of sixteen cities throughout North America.

[tex]
\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{8}{|c|}{\text{Temperatures (in } ^{\circ} F\text{)}} \\
\hline
93 & 105 & 110 & 107 & 103 & 106 & 94 & 102 \\
\hline
99 & 101 & 110 & 108 & 109 & 94 & 104 & 112 \\
\hline
\end{array}
\]
[/tex]

(a) Complete the grouped frequency distribution for the data. (Note that the class width is 4.)

[tex]
\[
\begin{array}{|c|c|}
\hline
\text{Temperatures (in } ^{\circ} F\text{)} & \text{Frequency} \\
\hline
92.5 \text{ to } 96.5 & 3 \\
96.5 \text{ to } 100.5 & 1 \\
100.5 \text{ to } 104.5 & 4 \\
104.5 \text{ to } 108.5 & 4 \\
108.5 \text{ to } 112.5 & \\
\hline
\end{array}
\]
[/tex]

(b) Using the classes from part (a), draw the frequency polygon for the data. Note that you can add or remove classes from the figure. Label each class with its midpoint.

Answer :

Sure! Let's go through the steps to complete the grouped frequency distribution and draw the frequency polygon for the hottest recorded temperatures in the sixteen cities.

### (a) Completing the Grouped Frequency Distribution

We have the hottest recorded temperatures:

- 93, 105, 110, 107, 103, 106, 94, 102,
- 99, 101, 110, 108, 109, 94, 104, 112

The task is to categorize these temperatures into groups (class intervals) with a class width of 4. The intervals provided are:

1. 92.5 to 96.5
2. 96.5 to 100.5
3. 100.5 to 104.5
4. 104.5 to 108.5
5. 108.5 to 112.5

Each of these class intervals has a specific range, and we need to count how many temperatures fall into each range.

- 92.5 to 96.5: This interval includes 93 and 94 (total of 3 temperatures).
- 96.5 to 100.5: This interval includes only 99 (total of 1 temperature).
- 100.5 to 104.5: This interval includes 103, 102, and 104 (total of 4 temperatures).
- 104.5 to 108.5: This interval includes 105, 107, 106, and 108 (total of 4 temperatures).
- 108.5 to 112.5: This interval includes 110, 110, 109, and 112 (total of 4 temperatures).

Here is the completed frequency distribution:

| Temperatures (in °F) | Frequency |
|----------------------|-----------|
| 92.5 to 96.5 | 3 |
| 96.5 to 100.5 | 1 |
| 100.5 to 104.5 | 4 |
| 104.5 to 108.5 | 4 |
| 108.5 to 112.5 | 4 |

### (b) Drawing the Frequency Polygon

To draw the frequency polygon, we need the midpoints of each class interval:

- Midpoint of 92.5 to 96.5: [tex]\( \frac{92.5 + 96.5}{2} = 94.5 \)[/tex]
- Midpoint of 96.5 to 100.5: [tex]\( \frac{96.5 + 100.5}{2} = 98.5 \)[/tex]
- Midpoint of 100.5 to 104.5: [tex]\( \frac{100.5 + 104.5}{2} = 102.5 \)[/tex]
- Midpoint of 104.5 to 108.5: [tex]\( \frac{104.5 + 108.5}{2} = 106.5 \)[/tex]
- Midpoint of 108.5 to 112.5: [tex]\( \frac{108.5 + 112.5}{2} = 110.5 \)[/tex]

Now plot these midpoints against their corresponding frequencies on a graph and connect the points with straight lines to form the frequency polygon. Make sure to label each class with its midpoint on the graph.

This completes the solution for the problem!