QUESTION IMAGE
Question
the data in the table represent the tuition for all 2 - year community colleges in a region in 2022 - 2023. (a) construct a cumulative frequency distribution. (b) construct a cumulative relative frequency distribution. tables with tuition (dollars) ranges (775 - 799, 800 - 824, 825 - 849, 850 - 874, 875 - 899, 900 - 924, 925 - 949, 950 - 974) and number of community colleges, and another table for cumulative relative frequency. (round to three decimal places as needed)
Part (a): Cumulative Frequency Distribution
First, we need to find the total number of community colleges. Let's sum up the frequencies: \(22 + 90 + 102 + 106 + 108 + 106 + 106 + 107\). Wait, actually, looking at the left table (maybe a typo, but the right table has the frequencies? Wait, the left table's "Number of Community Colleges" seems to have values: 22, 90, 102, 106, 108, 106, 106, 107? Wait, no, the left table's rows: 775 - 799: 22; 800 - 824: 90; 825 - 849: 102; 850 - 874: 106; 875 - 899: 108; 900 - 924: 106; 925 - 949: 106; 950 - 974: 107. Let's confirm the total:
\(22 + 90 = 112\); \(112 + 102 = 214\); \(214 + 106 = 320\); \(320 + 108 = 428\); \(428 + 106 = 534\); \(534 + 106 = 640\); \(640 + 107 = 747\). So total \(N = 747\).
Now, cumulative frequency is the sum of frequencies up to that class.
- For 775 - 799: Cumulative Frequency (CF) = 22
- For 800 - 824: CF = 22 + 90 = 112
- For 825 - 849: CF = 112 + 102 = 214
- For 850 - 874: CF = 214 + 106 = 320
- For 875 - 899: CF = 320 + 108 = 428
- For 900 - 924: CF = 428 + 106 = 534
- For 925 - 949: CF = 534 + 106 = 640
- For 950 - 974: CF = 640 + 107 = 747
Part (b): Cumulative Relative Frequency Distribution
Cumulative Relative Frequency (CRF) is \(\frac{\text{Cumulative Frequency}}{N}\), where \(N = 747\).
- For 775 - 799: \(CRF = \frac{22}{747} \approx 0.03\) (rounded to two decimal places)
- For 800 - 824: \(CRF = \frac{112}{747} \approx 0.15\)
- For 825 - 849: \(CRF = \frac{214}{747} \approx 0.29\)
- For 850 - 874: \(CRF = \frac{320}{747} \approx 0.43\)
- For 875 - 899: \(CRF = \frac{428}{747} \approx 0.57\)
- For 900 - 924: \(CRF = \frac{534}{747} \approx 0.71\)
- For 925 - 949: \(CRF = \frac{640}{747} \approx 0.86\)
- For 950 - 974: \(CRF = \frac{747}{747} = 1.00\)
Final Answers (for Cumulative Relative Frequency):
| Tuition (dollars) | Cumulative Relative Frequency |
|---|---|
| 800 - 824 | \( \approx 0.15 \) |
| 825 - 849 | \( \approx 0.29 \) |
| 850 - 874 | \( \approx 0.43 \) |
| 875 - 899 | \( \approx 0.57 \) |
| 900 - 924 | \( \approx 0.71 \) |
| 925 - 949 | \( \approx 0.86 \) |
| 950 - 974 | \( 1.00 \) |
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Part (a): Cumulative Frequency Distribution
First, we need to find the total number of community colleges. Let's sum up the frequencies: \(22 + 90 + 102 + 106 + 108 + 106 + 106 + 107\). Wait, actually, looking at the left table (maybe a typo, but the right table has the frequencies? Wait, the left table's "Number of Community Colleges" seems to have values: 22, 90, 102, 106, 108, 106, 106, 107? Wait, no, the left table's rows: 775 - 799: 22; 800 - 824: 90; 825 - 849: 102; 850 - 874: 106; 875 - 899: 108; 900 - 924: 106; 925 - 949: 106; 950 - 974: 107. Let's confirm the total:
\(22 + 90 = 112\); \(112 + 102 = 214\); \(214 + 106 = 320\); \(320 + 108 = 428\); \(428 + 106 = 534\); \(534 + 106 = 640\); \(640 + 107 = 747\). So total \(N = 747\).
Now, cumulative frequency is the sum of frequencies up to that class.
- For 775 - 799: Cumulative Frequency (CF) = 22
- For 800 - 824: CF = 22 + 90 = 112
- For 825 - 849: CF = 112 + 102 = 214
- For 850 - 874: CF = 214 + 106 = 320
- For 875 - 899: CF = 320 + 108 = 428
- For 900 - 924: CF = 428 + 106 = 534
- For 925 - 949: CF = 534 + 106 = 640
- For 950 - 974: CF = 640 + 107 = 747
Part (b): Cumulative Relative Frequency Distribution
Cumulative Relative Frequency (CRF) is \(\frac{\text{Cumulative Frequency}}{N}\), where \(N = 747\).
- For 775 - 799: \(CRF = \frac{22}{747} \approx 0.03\) (rounded to two decimal places)
- For 800 - 824: \(CRF = \frac{112}{747} \approx 0.15\)
- For 825 - 849: \(CRF = \frac{214}{747} \approx 0.29\)
- For 850 - 874: \(CRF = \frac{320}{747} \approx 0.43\)
- For 875 - 899: \(CRF = \frac{428}{747} \approx 0.57\)
- For 900 - 924: \(CRF = \frac{534}{747} \approx 0.71\)
- For 925 - 949: \(CRF = \frac{640}{747} \approx 0.86\)
- For 950 - 974: \(CRF = \frac{747}{747} = 1.00\)
Final Answers (for Cumulative Relative Frequency):
| Tuition (dollars) | Cumulative Relative Frequency |
|---|---|
| 800 - 824 | \( \approx 0.15 \) |
| 825 - 849 | \( \approx 0.29 \) |
| 850 - 874 | \( \approx 0.43 \) |
| 875 - 899 | \( \approx 0.57 \) |
| 900 - 924 | \( \approx 0.71 \) |
| 925 - 949 | \( \approx 0.86 \) |
| 950 - 974 | \( 1.00 \) |