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

\begin{tabular}{|c|c|c|c|c|}
\hline \multicolumn{5}{|c|}{Temperatures (in [tex]${ }^{\circ} F$[/tex])} \\
\hline 105 & 111 & 109 & 117 & 113 \\
\hline 103 & 114 & 109 & 98 & 116 \\
\hline 109 & 104 & 107 & 115 & 102 \\
\hline
\end{tabular}

(a) Complete the grouped frequency distribution for the data. (Note that the class width is 5.) Construct a histogram for the data.

\begin{tabular}{|l|l|}
\hline
Temperatures (in [tex]${ }^{\circ} F$[/tex]) & Frequency \\
\hline
98 to 102 & \\
103 to 107 & \\
108 to 112 & \\
113 to 117 & \\
\hline
\end{tabular}

Certainly! I'll guide you step by step to complete the grouped frequency distribution for the given temperature data and construct a histogram:

### Grouped Frequency Distribution
Here we are given the hottest recorded temperatures (in Fahrenheit) for fifteen cities. The temperatures were:

[tex]\[ 105, 111, 109, 117, 113, 103, 114, 109, 98, 116, 109, 104, 107, 115, 102 \][/tex]

We will classify these temperatures into specified ranges with a class width of 5:

1. 98-102: This range includes the temperatures 98, 99, 100, 101, and 102.
2. 103-107: This range includes the temperatures 103, 104, 105, 106, and 107.
3. 108-112: This range includes the temperatures 108, 109, 110, 111, and 112.
4. 113-117: This range includes the temperatures 113, 114, 115, 116, and 117.

Now, let’s determine the frequency for each range:

1. 98-102:
- The temperatures in this range are: 98, 102.
- Frequency: 2

2. 103-107:
- The temperatures in this range are: 103, 104, 105, 107.
- Frequency: 4

3. 108-112:
- The temperatures in this range are: 109, 109, 109, 111.
- Frequency: 4

4. 113-117:
- The temperatures in this range are: 113, 114, 115, 116, 117.
- Frequency: 5

So, the grouped frequency distribution is as follows:

| Temperature (°F) | Frequency |
|------------------|-----------|
| 98-102 | 2 |
| 103-107 | 4 |
| 108-112 | 4 |
| 113-117 | 5 |

### Histogram Construction

To visually represent this grouped frequency distribution, we can construct a histogram:

1. Horizontal Axis (X-axis): Represent the temperature ranges.
2. Vertical Axis (Y-axis): Represent the frequencies.

Here's a simple outline of the histogram:

[tex]\[ \begin{array}{l|l|l|l|l|l|l|l} \text{Frequencies} \quad&\vdots & 5 &\bullet\bullet\bullet\bullet\bullet \\ \text{(Y-axis)}\quad&\vdots & 4 &\bullet\bullet\bullet\bullet & \bullet\bullet\bullet\bullet \\ \quad &\vdots & 3 &\bullet\bullet\bullet & \bullet\bullet\bullet \\ \quad &\vdots & 2 &\bullet\bullet & \bullet\bullet \\ \quad &\vdots & 1 &\bullet\bullet &\bullet \quad \quad&0& &98-102 \quad &103-107 \quad &108-112 \quad&113-117 \quad \end{array} \][/tex]

This histogram indicates:
- 2 cities recorded temperatures between 98-102 °F.
- 4 cities recorded temperatures between 103-107 °F.
- 4 cities recorded temperatures between 108-112 °F.
- 5 cities recorded temperatures between 113-117 °F.