the data in the table represent the tuition for all 2-year community colleges in a region in 2022-2023. (a) draw a frequency ogive. (b) draw a relative frequency ogive.

| tuition (dollars) | number of community colleges |
|-------------------|------------------------------|
| 775-799 | 20 |
| 800-824 | 68 |
| 825-849 | 12 |
| 850-874 | 5 |
| 875-899 | 0 |
| 900-924 | 0 |
| 925-949 | 0 |
| 950-974 | 2 |

(a) which of the following graphs is the frequency ogive?
○ a. ○ b.
graphs with cumulative freq vs tuition (774 to 974) and cumulative freq up to 120
○ c. ○ d.

Question

the data in the table represent the tuition for all 2-year community colleges in a region in 2022-2023. (a) draw a frequency ogive. (b) draw a relative frequency ogive.

| tuition (dollars) | number of community colleges |
|-------------------|------------------------------|
| 775-799           | 20                           |
| 800-824           | 68                           |
| 825-849           | 12                           |
| 850-874           | 5                            |
| 875-899           | 0                            |
| 900-924           | 0                            |
| 925-949           | 0                            |
| 950-974           | 2                            |

(a) which of the following graphs is the frequency ogive?
○ a.  ○ b.
graphs with cumulative freq vs tuition (774 to 974) and cumulative freq up to 120
○ c.  ○ d.

Accepted Answer

### Part (a) - Drawing Frequency Ogive
#### Step 1: Understand Frequency Ogive
A frequency ogive (or cumulative frequency graph) is a line graph that represents the cumulative frequency of a dataset. For grouped data (like tuition ranges here), we plot the upper class boundaries on the x - axis and the cumulative frequency on the y - axis.

#### Step 2: Determine Class Boundaries
- For the class $775 - 799$, the upper class boundary is $799.5$ (since the next class starts at $800$, we take the mid - point between $799$ and $800$).
- For the class $800 - 824$, the upper class boundary is $824.5$.
- For the class $825 - 849$, the upper class boundary is $849.5$.
- For the class $850 - 874$, the upper class boundary is $874.5$.
- For the class $875 - 899$, the upper class boundary is $899.5$.
- For the class $900 - 924$, the upper class boundary is $924.5$.
- For the class $925 - 949$, the upper class boundary is $949.5$.
- For the class $950 - 974$, the upper class boundary is $974.5$.

#### Step 3: Calculate Cumulative Frequency
- For the first class ($775 - 799$): Cumulative Frequency ($CF$) = $20$ (since it's the first class, cumulative frequency is the frequency of the class itself).
- For the second class ($800 - 824$): $CF=20 + 68=88$
- For the third class ($825 - 849$): $CF = 88+12 = 100$
- For the fourth class ($850 - 874$): $CF=100 + 5=105$
- For the fifth class ($875 - 899$): $CF = 105+0 = 105$
- For the sixth class ($900 - 924$): $CF=105 + 0=105$
- For the seventh class ($925 - 949$): $CF = 105+0 = 105$
- For the eighth class ($950 - 974$): $CF=105 + 2=107$

#### Step 4: Plot the Points
We plot the points $(799.5,20)$, $(824.5,88)$, $(849.5,100)$, $(874.5,105)$, $(899.5,105)$, $(924.5,105)$, $(949.5,105)$, $(974.5,107)$ and connect them with straight lines.

### Part (b) - Drawing Relative Frequency Ogive
#### Step 1: Understand Relative Frequency Ogive
A relative frequency ogive uses cumulative relative frequency. The relative frequency of a class is $\frac{\text{Frequency of the class}}{\text{Total number of observations}}$, and cumulative relative frequency is the sum of relative frequencies up to that class.

#### Step 2: Calculate Total Number of Community Colleges
Total number of community colleges $N=20 + 68+12 + 5+0 + 0+0 + 2=107$

#### Step 3: Calculate Relative Frequencies and Cumulative Relative Frequencies
- For the class $775 - 799$: Relative Frequency ($RF$)=$\frac{20}{107}\approx0.187$, Cumulative Relative Frequency ($CRF$)=$0.187$
- For the class $800 - 824$: $RF=\frac{68}{107}\approx0.636$, $CRF = 0.187+0.636 = 0.823$
- For the class $825 - 849$: $RF=\frac{12}{107}\approx0.112$, $CRF=0.823 + 0.112=0.935$
- For the class $850 - 874$: $RF=\frac{5}{107}\approx0.047$, $CRF=0.935+0.047 = 0.982$
- For the class $875 - 899$: $RF=\frac{0}{107}=0$, $CRF = 0.982+0 = 0.982$
- For the class $900 - 924$: $RF=\frac{0}{107}=0$, $CRF=0.982 + 0=0.982$
- For the class $925 - 949$: $RF=\frac{0}{107}=0$, $CRF=0.982+0 = 0.982$
- For the class $950 - 974$: $RF=\frac{2}{107}\approx0.019$, $CRF=0.982+0.019 = 1.001$ (due to rounding)

#### Step 4: Plot the Points
We plot the points $(799.5,0.187)$, $(824.5,0.823)$, $(849.5,0.935)$, $(874.5,0.982)$, $(899.5,0.982)$, $(924.5,0.982)$, $(949.5,0.982)$, $(974.5,1.001)$ and connect them with straight lines.

### Part (a) - Identifying the Frequency Ogive
A frequency ogive should be a non - decreasing (non - falling) graph since cumulative frequency never decreases. Let's analyze the options:
- Option A: The graph is decreasing, which is not possible for a cumulative frequency ogive.
- Option B: The graph has a peak and then decreases, which is not a cumulative frequency ogive (cumulative frequency should be non - decreasing).
- Option C: The graph has a very sharp increase at the end, which does not match our cumulative frequency values (our cumulative frequency increases gradually and then stays constant for some classes).
- Option D: The graph is non - decreasing. It starts low, has a significant increase (corresponding to the class with frequency 68), and then increases gradually and finally has a small increase. This matches the cumulative frequency pattern we calculated (since after the class with frequency 68, the cumulative frequency increases by smaller amounts and then stays constant for some classes before the last small increase).

# Answer (for part (a) - identifying the frequency ogive):
D

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QUESTION IMAGE

Question

Explanation:

Part (a) - Drawing Frequency Ogive

Step 1: Understand Frequency Ogive

Step 2: Determine Class Boundaries

Step 3: Calculate Cumulative Frequency

Step 4: Plot the Points

Part (b) - Drawing Relative Frequency Ogive

Step 1: Understand Relative Frequency Ogive

Step 2: Calculate Total Number of Community Colleges

Step 3: Calculate Relative Frequencies and Cumulative Relative Frequencies

Step 4: Plot the Points

Part (a) - Identifying the Frequency Ogive

Answer: