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