Questions: Time Spent Looking for a Parking Space The following frequency distribution shows the amount of time (in hours) that workers in a large city spend each year trying to find a parking space. Draw a histogram, frequency polygon, and ogive for the data. Summarize the results. Time Frequency 27-31 23 32-36 28 37-41 19 42-46 15 47-51 11 52-56 4/100

Time Spent Looking for a Parking Space The following frequency distribution shows the amount of time (in hours) that workers in a large city spend each year trying to find a parking space. Draw a histogram, frequency polygon, and ogive for the data. Summarize the results.
Time Frequency
27-31 23
32-36 28
37-41 19
42-46 15
47-51 11
52-56 4/100

Transcript text: Time Spent Looking for a Parking Space The following frequency distribution shows the amount of time (in hours) that workers in a large city spend each year trying to find a parking space. Draw a histogram, frequency polygon, and ogive for the data. Summarize the results. Time & Frequency $27-31$ & 23 $32-36$ & 28 $37-41$ & 19 $42-46$ & 15 $47-51$ & 11 $52-56$ & $\frac{4}{100}$

Solution

Solution Steps

Step 1: Calculate the Midpoints of Each Class Interval

To create the frequency polygon, we first need to calculate the midpoints of each class interval.

\[ \text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2} \]

\[ \begin{aligned} &\text{Midpoint of } 27-31 = \frac{27 + 31}{2} = 29 \\ &\text{Midpoint of } 32-36 = \frac{32 + 36}{2} = 34 \\ &\text{Midpoint of } 37-41 = \frac{37 + 41}{2} = 39 \\ &\text{Midpoint of } 42-46 = \frac{42 + 46}{2} = 44 \\ &\text{Midpoint of } 47-51 = \frac{47 + 51}{2} = 49 \\ &\text{Midpoint of } 52-56 = \frac{52 + 56}{2} = 54 \\ \end{aligned} \]

Step 2: Create the Frequency Polygon Data

We use the midpoints and frequencies to create the frequency polygon.

\[ \begin{aligned} &\text{(29, 23)} \\ &\text{(34, 28)} \\ &\text{(39, 19)} \\ &\text{(44, 15)} \\ &\text{(49, 11)} \\ &\text{(54, 0.04)} \\ \end{aligned} \]

Step 3: Create the Ogive Data

To create the ogive, we need cumulative frequencies.

\[ \begin{aligned} &\text{Cumulative frequency at 31} = 23 \\ &\text{Cumulative frequency at 36} = 23 + 28 = 51 \\ &\text{Cumulative frequency at 41} = 51 + 19 = 70 \\ &\text{Cumulative frequency at 46} = 70 + 15 = 85 \\ &\text{Cumulative frequency at 51} = 85 + 11 = 96 \\ &\text{Cumulative frequency at 56} = 96 + 0.04 = 96.04 \\ \end{aligned} \]

Final Answer

The midpoints and frequencies for the frequency polygon are: \[ \begin{aligned} &\text{(29, 23)} \\ &\text{(34, 28)} \\ &\text{(39, 19)} \\ &\text{(44, 15)} \\ &\text{(49, 11)} \\ &\text{(54, 0.04)} \\ \end{aligned} \]

The cumulative frequencies for the ogive are: \[ \begin{aligned} &\text{(31, 23)} \\ &\text{(36, 51)} \\ &\text{(41, 70)} \\ &\text{(46, 85)} \\ &\text{(51, 96)} \\ &\text{(56, 96.04)} \\ \end{aligned} \]

{"axisType": 3, "coordSystem": {"xmin": 25, "xmax": 60, "ymin": 0, "ymax": 100}, "commands": ["y = 23 if 27 <= x <= 31 else 0", "y = 28 if 32 <= x <= 36 else 0", "y = 19 if 37 <= x <= 41 else 0", "y = 15 if 42 <= x <= 46 else 0", "y = 11 if 47 <= x <= 51 else 0", "y = 0.04 if 52 <= x <= 56 else 0"], "latex_expressions": ["$y = 23$ for $27 \\leq x \\leq 31$", "$y = 28$ for $32 \\leq x \\leq 36$", "$y = 19$ for $37 \\leq x \\leq 41$", "$y = 15$ for $42 \\leq x \\leq 46$", "$y = 11$ for $47 \\leq x \\leq 51$", "$y = 0.04$ for $52 \\leq x \\leq 56$"]}

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