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.

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [],
"new_concepts": [
"Frequency Polygon",
"Class Midpoint",
"Class Boundaries"
],
"current_concepts": [
"Frequency Polygon",
"Class Midpoint",
"Class Boundaries"
]
}
</pre_analysis>

<reasoning>

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:
\[
(14, 0), (19, 8), (24, 3), (29, 2), (34, 7), (39, 2), (44, 0)
\]
</reasoning>

<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.
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[14, 0]],
"properties": {
"name": "(14,0)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[19, 8]],
"properties": {
"name": "(19,8)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[24, 3]],
"properties": {
"name": "(24,3)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[29, 2]],
"properties": {
"name": "(29,2)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[34, 7]],
"properties": {
"name": "(34,7)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[39, 2]],
"properties": {
"name": "(39,2)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[44, 0]],
"properties": {
"name": "(44,0)",
"size": 3,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "segment",
"params": [[14, 0], [19, 8]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[19, 8], [24, 3]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[24, 3], [29, 2]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[29, 2], [34, 7]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[34, 7], [39, 2]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "segment",
"params": [[39, 2], [44, 0]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Statistics",
"Frequency Polygon"
]
}
</post_analysis>