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the data in the table represent the tuition for all 2-year community co…

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
800-82468
825-84912
850-8745
875-8990
900-9240
925-9490
950-9742

(b) which of the following graphs is the relative frequency ogive?
options a, b, c, d with graphs

Explanation:

Response
Part (a): Drawing a Frequency Ogive

A frequency ogive (cumulative frequency graph) is constructed by plotting cumulative frequency against the upper class boundaries of each interval.

Step 1: Determine Class Boundaries

For a class like \( 775 - 799 \), the lower boundary is \( 774.5 \) (since \( 775 - 0.5 = 774.5 \)) and the upper boundary is \( 799.5 \) (since \( 799 + 0.5 = 799.5 \)). Similarly:

  • \( 775 - 799 \): Boundaries \( 774.5 - 799.5 \)
  • \( 800 - 824 \): \( 799.5 - 824.5 \)
  • \( 825 - 849 \): \( 824.5 - 849.5 \)
  • \( 850 - 874 \): \( 849.5 - 874.5 \)
  • \( 875 - 899 \): \( 874.5 - 899.5 \)
  • \( 900 - 924 \): \( 899.5 - 924.5 \)
  • \( 925 - 949 \): \( 924.5 - 949.5 \)
  • \( 950 - 974 \): \( 949.5 - 974.5 \)
Step 2: Calculate Cumulative Frequency

Cumulative frequency is the sum of frequencies up to each class.

  • \( 775 - 799 \): \( 20 \) (cumulative: \( 20 \))
  • \( 800 - 824 \): \( 68 \) (cumulative: \( 20 + 68 = 88 \))
  • \( 825 - 849 \): \( 12 \) (cumulative: \( 88 + 12 = 100 \))
  • \( 850 - 874 \): \( 5 \) (cumulative: \( 100 + 5 = 105 \))
  • \( 875 - 899 \): \( 0 \) (cumulative: \( 105 + 0 = 105 \))
  • \( 900 - 924 \): \( 0 \) (cumulative: \( 105 + 0 = 105 \))
  • \( 925 - 949 \): \( 0 \) (cumulative: \( 105 + 0 = 105 \))
  • \( 950 - 974 \): \( 2 \) (cumulative: \( 105 + 2 = 107 \))
Step 3: Plot the Ogive
  • On the x - axis, mark the upper class boundaries (\( 799.5, 824.5, 849.5, 874.5, 899.5, 924.5, 949.5, 974.5 \)).
  • On the y - axis, mark the cumulative frequencies (\( 20, 88, 100, 105, 105, 105, 105, 107 \)).
  • Plot points at (upper boundary, cumulative frequency) and connect them with straight lines. Start from the lower boundary of the first class (\( 774.5 \)) with cumulative frequency \( 0 \).
Part (b): Identifying the Relative Frequency Ogive

A relative frequency ogive plots cumulative relative frequency (cumulative frequency / total number of observations) against upper class boundaries.

Step 1: Calculate Total Number of Colleges

Sum all frequencies: \( 20 + 68 + 12 + 5 + 0 + 0 + 0 + 2 = 107 \).

Step 2: Calculate Cumulative Relative Frequency
  • \( 775 - 799 \): \( \frac{20}{107} \approx 0.187 \) (cumulative: \( 0.187 \))
  • \( 800 - 824 \): \( \frac{68}{107} \approx 0.636 \) (cumulative: \( 0.187 + 0.636 \approx 0.823 \))
  • \( 825 - 849 \): \( \frac{12}{107} \approx 0.112 \) (cumulative: \( 0.823 + 0.112 \approx 0.935 \))
  • \( 850 - 874 \): \( \frac{5}{107} \approx 0.047 \) (cumulative: \( 0.935 + 0.047 \approx 0.982 \))
  • \( 875 - 899 \): \( 0 \) (cumulative: \( 0.982 + 0 = 0.982 \))
  • \( 900 - 924 \): \( 0 \) (cumulative: \( 0.982 + 0 = 0.982 \))
  • \( 925 - 949 \): \( 0 \) (cumulative: \( 0.982 + 0 = 0.982 \))
  • \( 950 - 974 \): \( \frac{2}{107} \approx 0.019 \) (cumulative: \( 0.982 + 0.019 \approx 1.001 \) (due to rounding))
Step 3: Analyze the Graphs

A relative frequency ogive should start at \( 0 \) (at the lower boundary of the first class, \( 774.5 \)) and increase to \( 1 \) (or near \( 1 \)) at the upper boundary of the last class. The graph should show cumulative growth, with most growth in the \( 800 - 824 \) interval (since it has the highest frequency). From the options, the graph with cumulative relative frequency increasing (starting at \( 0 \), rising steadily, especially in the \( 800 - 824 \) range, and approaching \( 1 \)) is the correct relative frequency ogive. Assuming the options are labeled, the correct graph will match the cumulative relative frequency pattern (e.g., if Option A shows a curve starting at \( 0 \), rising with a steep increase in the…

Answer:

(a) [Frequency ogive plotted as described]
(b) A (or the specific option matching the cumulative relative frequency pattern)