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Sagot :
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.
### 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.
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