91Ó°ÊÓ

Use the specified number of classes to do the following. (a) Find the class width. (b) Make a frequency table showing class limits, class boundaries, midpoints, frequencies, relative frequencies, and cumulative frequencies. (c) Draw a histogram. (d) Draw a relative-frequency histogram. (e) Categorize the basic distribution shape as uniform, mound-shaped symmetrical, bimodal, skewed left, or skewed right. (f) Draw an ogive. How long does it take to finish the 1161 -mile Iditarod Dog Sled Race from Anchorage to Nome, Alaska (see Viewpoint)? Finish times (to the nearest hour) for 57 dogsled teams are shown below. $$ \begin{array}{llllllllllll} 261 & 271 & 236 & 244 & 279 & 296 & 284 & 299 & 288 & 288 & 247 & 256 \\ 338 & 360 & 341 & 333 & 261 & 266 & 287 & 296 & 313 & 311 & 307 & 307 \\ 299 & 303 & 277 & 283 & 304 & 305 & 288 & 290 & 288 & 289 & 297 & 299 \\ 332 & 330 & 309 & 328 & 307 & 328 & 285 & 291 & 295 & 298 & 306 & 315 \\ 310 & 318 & 318 & 320 & 333 & 321 & 323 & 324 & 327 & & & \end{array} $$ Use five classes.

Step by step solution

01

Identify the Range and Determine Class Width

First, find the range by subtracting the smallest value from the largest value:The smallest finish time is 236 hours and the largest is 360 hours. Therefore, the range is:\[\text{Range} = 360 - 236 = 124\]\Now divide the range by the number of classes, which is 5, to find the class width:\[\text{Class Width} = \frac{124}{5} = 24.8\]Round this number up to the next whole number to ensure all data fits within the class limits. So, the class width is 25.

02

Construct Class Limits

Start with the smallest data point as the lower class limit of the first class. Include five classes and use the class width to establish class limits: - Class 1: 236 - 260 - Class 2: 261 - 285 - Class 3: 286 - 310 - Class 4: 311 - 335 - Class 5: 336 - 360

03

Determine Class Boundaries and Midpoints

The class boundaries account for the space between classes: - Class 1: 235.5 - 260.5 - Class 2: 260.5 - 285.5 - Class 3: 285.5 - 310.5 - Class 4: 310.5 - 335.5 - Class 5: 335.5 - 360.5 The midpoint of each class is found by averaging the upper and lower class limits: - Class 1 Midpoint: (236 + 260) / 2 = 248 - Class 2 Midpoint: (261 + 285) / 2 = 273 - Class 3 Midpoint: (286 + 310) / 2 = 298 - Class 4 Midpoint: (311 + 335) / 2 = 323 - Class 5 Midpoint: (336 + 360) / 2 = 348

04

Frequency, Relative Frequency, and Cumulative Frequency

Count the number of data points within each class interval: - Class 1: 4 - Class 2: 14 - Class 3: 19 - Class 4: 11 - Class 5: 9 Calculate relative frequency (frequency divided by the total number of observations, 57): - Class 1: 4/57 ~ 0.07 - Class 2: 14/57 ~ 0.25 - Class 3: 19/57 ~ 0.33 - Class 4: 11/57 ~ 0.19 - Class 5: 9/57 ~ 0.16 Cumulative frequency is obtained by sequentially adding the frequencies: - Class 1: 4 - Class 2: 18 - Class 3: 37 - Class 4: 48 - Class 5: 57

05

Draw Histogram

Using the class limits and frequencies, draw bars for each class. The x-axis represents the class intervals, and the y-axis represents the frequency counts. Classes will appear as 236-260, 261-285, etc., with the respective frequency heights. Each bar's left edge aligns with the lower class limit, and the right edge aligns with the upper class limit.

06

Draw Relative Frequency Histogram

This histogram substitutes class frequencies with relative frequencies. The x-axis lists class intervals, and the y-axis corresponds to relative frequencies. Draw each bar based on relative frequency values from Step 4: 0.07, 0.25, 0.33, 0.19, and 0.16.

07

Analyze Distribution Shape

Examine the histogram's shape. Given most data lies in the middle classes with comparatively fewer in extreme classes, the distribution is mound-shaped symmetrical, indicating a normal distribution shape.

08

Draw an Ogive

Plot an ogive graph with the cumulative frequencies along the y-axis and class boundaries along the x-axis. Begin plotting at the lower boundary of the first class and mark cumulative frequency at the upper boundary of each class. Connect the dots with a continuous line to display cumulative frequency surge across intervals.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Table

A frequency table is a useful tool for summarizing and organizing data. It helps in making complex data easier to understand by tallying how often each observation occurs in different data categories. In the context of the Iditarod Dog Sled Race times, the data is organized into five classes based on the finishing times of the teams.
To create a frequency table, follow these steps:

• Identify the range of data by subtracting the smallest data point from the largest.
• Determine the class width using the range divided by the number of desired classes, rounding up as necessary (in this case, 25).
• Set class limits to ensure all data points fit within them.
• Determine class boundaries, which are slightly adjusted limits to account for any gaps between classes.
• Calculate class midpoints by averaging the upper and lower class limits.
• Count the number of observations in each class to find the frequency.
• Find relative frequency by dividing the class frequency by the total observations. Cumulative frequency is found by adding the frequencies sequentially through the classes.

A frequency table provides a structured view of data that’s essential for further statistical analysis.

Histogram

A histogram is a type of bar graph used to visualize the distribution of a dataset. It graphically depicts the frequency distribution by showing how data is spread across different intervals, or bins. For the Iditarod Dog Sled Race data, we need to visualize this information to gain insights into the pattern of finish times.
When creating a histogram:

• Use the frequency table as a basis, plotting the distinct class intervals on the horizontal axis (x-axis).
• Mark the frequency on the vertical axis (y-axis).
• Each bar in the histogram represents the frequency of data within each class interval, drawn without gaps between bars to emphasize continuity.

This visual representation helps to quickly understand how data points are distributed across different classes. By looking at the produced histogram, you can see patterns, such as clustering of data, and identify characteristics like skewness or symmetry.

Ogive

An ogive is a graph that represents the cumulative frequency or cumulative relative frequency of a dataset. It provides a way to understand how frequencies accumulate over different intervals, showcasing cumulative progressions.
To draw an ogive for the Iditarod data:

• Utilize the cumulative frequency from the frequency table.
• Plot cumulative frequencies at the upper boundary of each class interval. Start at zero at the lower boundary of the first class.
• Connect these plotted points with a smooth line.
  

An ogive is particularly beneficial when determining medians, percentiles, and other statistical measurements. It provides a long-term view of the data’s cumulative frequency, showing how quickly or slowly data accumulates as it moves across various intervals.

Distribution Shape

The shape of a dataset's distribution gives you insight into the data's overall patterns and tendencies. Analyzing the shape involves examining the histogram for the Iditarod race times and observing how data points are arranged.
Different distribution shapes include:

• Uniform: Similar frequency across all intervals, forming a box-like shape.
• Mound-shaped Symmetrical: A bell-shaped curve that indicates a normal distribution, where most data clusters around the center.
• Bimodal: Two peaks in the data indicate distinct modes within the dataset.
• Skewed Left: Distribution tails off to the left, indicating a concentration of higher values.
• Skewed Right: Distribution tails off to the right, showing a concentration of lower values.

The Iditarod data, when inspected through its histogram, may exhibit a mound-shaped symmetrical distribution, suggesting a normal distribution pattern. Understanding these shapes is key to making predictions and comprehending underlying processes in the dataset.