Question

a sample of 22 participants took part in a hearing experiment. among other things, the absolute hearing threshold (in decibels) was measured for each participant. the 22 measurements were as follows:

25, 22, 18, 35, 37, 34, 31, 27, 24, 21, 18, 39, 20, 20, 20, 17, 17, 32, 32, 35, 35, 35

draw the frequency polygon for these data using an initial class boundary of 16.5 and a class width of 5. note that you can add or remove classes from the figure. label each class with its midpoint.

Explanation:

Determine the classes, midpoints, and frequencies

Given data (22 measurements):
\(25, 22, 18, 35, 37, 34, 31, 27, 24, 21, 18, 39, 20, 20, 20, 17, 17, 32, 32, 35, 35, 35\)

Sorted data:
\(17, 17, 18, 18, 20, 20, 20, 21, 22, 24, 25, 27, 31, 32, 32, 34, 35, 35, 35, 35, 37, 39\)

Class boundaries starting at \(16.5\) with width \(5\):

• Class 1: \(16.5 - 21.5\)
• Data points: \(17, 17, 18, 18, 20, 20, 20, 21\) (8 values)
• Midpoint: \(\frac{16.5 + 21.5}{2} = 19\)
• Frequency: \(8\)
• Class 2: \(21.5 - 26.5\)
• Data points: \(22, 24, 25\) (3 values)
• Midpoint: \(\frac{21.5 + 26.5}{2} = 24\)
• Frequency: \(3\)
• Class 3: \(26.5 - 31.5\)
• Data points: \(27, 31\) (2 values)
• Midpoint: \(\frac{26.5 + 31.5}{2} = 29\)
• Frequency: \(2\)
• Class 4: \(31.5 - 36.5\)
• Data points: \(32, 32, 34, 35, 35, 35, 35\) (7 values)
• Midpoint: \(\frac{31.5 + 36.5}{2} = 34\)
• Frequency: \(7\)
• Class 5: \(36.5 - 41.5\)
• Data points: \(37, 39\) (2 values)
• Midpoint: \(\frac{36.5 + 41.5}{2} = 39\)
• Frequency: \(2\)

Anchor the frequency polygon to zero

To complete the frequency polygon, anchor it to the horizontal axis (frequency of 0) at the midpoints of the immediate preceding and succeeding classes:

• Preceding anchor midpoint: \(19 - 5 = 14\) (Frequency: \(0\))
• Succeeding anchor midpoint: \(39 + 5 = 44\) (Frequency: \(0\))

Summarize the coordinates for plotting

The points to plot and connect with straight lines are:

$$ (14, 0), (19, 8), (24, 3), (29, 2), (34, 7), (39, 2), (44, 0) $$

Answer:

To construct the frequency polygon, use the following class midpoints and frequencies, anchored to zero at both ends:

Class Boundaries	Midpoint	Frequency	Coordinate to Plot
\(16.5 - 21.5\)	\(19\)	\(8\)	\((19, 8)\)
\(21.5 - 26.5\)	\(24\)	\(3\)	\((24, 3)\)
\(26.5 - 31.5\)	\(29\)	\(2\)	\((29, 2)\)
\(31.5 - 36.5\)	\(34\)	\(7\)	\((34, 7)\)
\(36.5 - 41.5\)	\(39\)	\(2\)	\((39, 2)\)
Anchor Class	\(44\)	\(0\)	\((44, 0)\)

Plot these points on your graph and connect them sequentially with straight line segments.