High School

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

[tex]\[
\begin{array}{|cccccccc|}
\hline
\multicolumn{8}{|c|}{\text{Temperatures (in } ^{\circ} F \text{)}} \\
\hline
105 & 110 & 97 & 103 & 112 & 94 & 94 & 106 \\
105 & 101 & 96 & 99 & 104 & 102 & 99 & 93 \\
\hline
\end{array}
\][/tex]

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

(b) Construct a histogram for the data.

[tex]\[
\begin{array}{|cc|}
\hline
\text{Temperatures (in } ^{\circ} F \text{)} & \text{Frequency} \\
\hline
92.5-96.5 & 3 \\
96.5-100.5 & 4 \\
100.5-104.5 & 4 \\
104.5-108.5 & 3 \\
108.5-112.5 & 2 \\
\hline
\end{array}
\][/tex]

Answer :

Certainly! Let's work through building a grouped frequency distribution and a histogram for the given data:

### Step-by-Step Solution:

#### (a) Complete the grouped frequency distribution for the data.

1. Data Set: We have the following temperatures:

105, 110, 97, 103, 112, 94, 94, 106, 105, 101, 96, 99, 104, 102, 99, 93.

2. Determine Class Intervals and Width:
- Given class width is 4.
- Use the midpoint of the classes to define ranges:
- (92.5, 96.5)
- (96.5, 100.5)
- (100.5, 104.5)
- (104.5, 108.5)
- (108.5, 112.5)

3. Calculate Frequency for Each Interval:
- Count how many temperatures fall into each interval.

- Interval (92.5, 96.5):
- Temperatures: 94, 94, 96, 93
- Frequency: 4

- Interval (96.5, 100.5):
- Temperatures: 97, 99, 99
- Frequency: 3

- Interval (100.5, 104.5):
- Temperatures: 101, 103, 104
- Frequency: 4

- Interval (104.5, 108.5):
- Temperatures: 105, 105, 106
- Frequency: 3

- Interval (108.5, 112.5):
- Temperatures: 110, 112
- Frequency: 2

4. Grouped Frequency Distribution Table:
- | Temperatures (in °F) | Frequency |
|----------------------|-----------|
| 92.5 - 96.5 | 4 |
| 96.5 - 100.5 | 3 |
| 100.5 - 104.5 | 4 |
| 104.5 - 108.5 | 3 |
| 108.5 - 112.5 | 2 |

#### (b) Construct a histogram for the data:

1. Create Histogram Representation:
- Make a bar for each interval on the x-axis.
- Height of each bar corresponds to the frequency of temperatures within that interval.

2. Visual Representation:
- Draw x-axis with intervals labeled: (92.5 - 96.5), (96.5 - 100.5), etc.
- Draw y-axis to represent frequency.
- Plot bars:
- (92.5 - 96.5): Bar with height of 4
- (96.5 - 100.5): Bar with height of 3
- (100.5 - 104.5): Bar with height of 4
- (104.5 - 108.5): Bar with height of 3
- (108.5 - 112.5): Bar with height of 2

By following these steps, you successfully complete the grouped frequency distribution and create a histogram for the temperature data.

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