Use the frequency distribution shown below to construct an expanded frequency distribution.
High Temperatures ​(degrees Upper F​)
Class
17​-27
28​-38
39​-49
50​-60
61​-71
72​-82
83​-93
 
​Frequency, f
18
42
68
69
78
68
22
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Part 1
Complete the table below. 
High Temperatures ​(degrees Upper F​)     ​(Round to the nearest hundredth as​ needed.)
Class
​Frequency, f
Midpoint
Relative frequency
Cumulative frequency
17​-27
18
  
enter your response here
  
enter your response here
  
enter your response here


We are tasked with constructing an expanded frequency distribution for the given frequency data, which includes calculating midpoints, relative frequencies, and cumulative frequencies.

Let’s break it down step by step.

### 1. **Midpoint Calculation:**
The midpoint for each class is the average of the lower and upper boundaries. The formula for the midpoint is:

\[
\text{Midpoint} = \frac{\text{Lower boundary} + \text{Upper boundary}}{2}
\]

### 2. **Relative Frequency Calculation:**
The relative frequency is the fraction or percentage of the total frequency for each class. The formula is:

\[
\text{Relative Frequency} = \frac{\text{Frequency of the class}}{\text{Total Frequency}}
\]

### 3. **Cumulative Frequency Calculation:**
Cumulative frequency is the running total of the frequencies up to the current class. 

Now, let's apply this to each class:

#### Step 1: Midpoints

For each class, we calculate the midpoint:

1. \( 17 - 27 \): \( \frac{17 + 27}{2} = 22 \)
2. \( 28 - 38 \): \( \frac{28 + 38}{2} = 33 \)
3. \( 39 - 49 \): \( \frac{39 + 49}{2} = 44 \)
4. \( 50 - 60 \): \( \frac{50 + 60}{2} = 55 \)
5. \( 61 - 71 \): \( \frac{61 + 71}{2} = 66 \)
6. \( 72 - 82 \): \( \frac{72 + 82}{2} = 77 \)
7. \( 83 - 93 \): \( \frac{83 + 93}{2} = 88 \)

#### Step 2: Relative Frequencies

First, sum the total frequencies: 

\[
18 + 42 + 68 + 69 + 78 + 68 + 22 = 365
\]

Now calculate the relative frequency for each class:

1. \( \frac{18}{365} \approx 0.0493 \)
2. \( \frac{42}{365} \approx 0.1151 \)
3. \( \frac{68}{365} \approx 0.1863 \)
4. \( \frac{69}{365} \approx 0.1890 \)
5. \( \frac{78}{365} \approx 0.2137 \)
6. \( \frac{68}{365} \approx 0.1863 \)
7. \( \frac{22}{365} \approx 0.0603 \)

#### Step 3: Cumulative Frequencies

Cumulative frequencies are calculated by adding the frequency of the current class to the sum of the frequencies of the previous classes:

1. \( 18 \)
2. \( 18 + 42 = 60 \)
3. \( 60 + 68 = 128 \)
4. \( 128 + 69 = 197 \)
5. \( 197 + 78 = 275 \)
6. \( 275 + 68 = 343 \)
7. \( 343 + 22 = 365 \)

Now we can fill in the table:

| High Temperatures (°F) | Frequency (f) | Midpoint | Relative Frequency | Cumulative Frequency |
|------------------------|---------------|----------|--------------------|----------------------|
| 17-27                  | 18            | 22       | 0.0493             | 18                   |
| 28-38                  | 42            | 33       | 0.1151             | 60                   |
| 39-49                  | 68            | 44       | 0.1863             | 128                  |
| 50-60

Use the frequency distribution shown below to construct an expanded frequency distribution. High Temperatures (degrees F) Class 17-27, 28-38, 39-49, 50-60, 61-71, 72-82, 83-93. Frequency, f: 18, 42, 68, 69, 78, 68, 22. Complete the table with Midpoint, Relative frequency, Cumulative frequency.

Learn how to construct an expanded frequency distribution for high temperatures ranging from 17°F to 93°F. This step-by-step guide covers calculating midpoints, relative frequencies, and cumulative frequencies using the provided data. Suitable for students in Grades 9-12.

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