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Fire load \(\left(\mathrm{MJ} / \mathrm{m}^{2}\right)\) is the heat energy that could be released per square meter of floor area by combustion of contents and the structure itself. The article "Fire Loads in Office Buildings" \((J .\) Struct. Engrg., 1997: \(365-368\) ) gave the following cumulative percentages (read from a graph) for fire loads in a sample of 388 rooms: \(\begin{array}{lccccc}\text { Value } & 0 & 150 & 300 & 450 & 600 \\ \text { Cumulative \% } & 0 & 19.3 & 37.6 & 62.7 & 77.5 \\ \text { Value } & 750 & 900 & 1050 & 1200 & 1350 \\ \text { Cumulative \% } & 87.2 & 93.8 & 95.7 & 98.6 & 99.1 \\ \text { Value } & 1500 & 1650 & 1800 & 1950 & \\ \text { Cumulative \% } & 99.5 & 99.6 & 99.8 & 100.0\end{array}\) a. Construct a relative frequency histogram and comment on interesting features. b. What proportion of fire loads are less than 600 ? At least 1200 ? c. What proportion of the loads are between 600 and 1200 ?

Step by step solution

01

Understand the Data

We have cumulative percentage data for fire loads at different values. This tells us the percentage of rooms with fire loads up to each specified value from a sample of 388 rooms.

02

Construct Relative Frequencies

To construct the relative frequency for each interval, subtract the cumulative percentage of the previous value from the current value, then divide by 100 to convert to a proportion. For example, the relative frequency for the interval 0 to 150 is \((19.3 - 0)/100 = 0.193\).

03

Draw the Relative Frequency Histogram

Using the relative frequencies calculated, draw a histogram where each bar height corresponds to the relative frequency of its interval. The base of each bar represents the fire load interval.

04

Identify Histogram Features

Upon examining the histogram, note any patterns such as skewness, peaks, or gaps. For this data, observe whether it is skewed left or right, or if there is a high frequency in specific intervals.

05

Calculate Proportions of Interest a (Less than 600)

The cumulative percentage for fire loads less than 600 is 77.5%. Convert this percentage to a proportion: \(77.5/100 = 0.775\).

06

Calculate Proportions of Interest b (At Least 1200)

To find the proportion of fire loads at least 1200, subtract the cumulative percentage of loads less than 1200 from 100%: \(100 - 98.6 = 1.4\). Convert this percentage to a proportion: \(1.4/100 = 0.014\).

07

Calculate Proportions of Interest c (Between 600 and 1200)

The cumulative percentage for less than 600 is 77.5%, and for less than 1200 it is 98.6%. Subtract the percentage for less than 600 from the percentage for less than 1200 to find the relative percentage: \(98.6 - 77.5 = 21.1\). The proportion is \(21.1/100 = 0.211\).

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Key Concepts

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

Relative Frequency Histogram

A relative frequency histogram is a type of bar chart that visually displays the relative frequencies of data intervals. Each bar's height represents the relative frequency of the interval. It's a great way to see how frequently different ranges of data occur within a dataset.

To create it, follow these steps:

• Identify data intervals, like 0-150, 150-300, etc.
• Compute the relative frequency for each interval. Subtract the cumulative percentage at the start of the interval from the end, then divide by 100. For instance, the interval 0-150 has a relative frequency of \(0.193\).
• Plot a bar for each interval where the height equals its relative frequency.

As you analyze the histogram, observe how the bars represent different data distributions. Look for patterns, such as peaks (where bars reach their maximum) or skewness (where most data is concentrated). It's an engaging method to summarize data quickly!

Cumulative Percentage

Cumulative percentage is a useful metric for understanding how data accumulates over intervals. It answers the question: "Up to this point, what portion of the total dataset is included?" This is particularly valuable for determining the spread and concentration of data points.

For example, a cumulative percentage of 62.7% at a 450 fire load indicates that 62.7% of rooms have fire loads up to 450 MJ/m².

Calculate it by adding the percentage found for each successive interval. This produces a "running total" of percentages. It's crucial for constructing cumulative frequency curves and determining significant benchmarks in datasets, such as medians or quartiles.

Knowing cumulative percentages also helps in comparing how two datasets accumulate. This can offer insights about differences in trends or distributions.

Proportion Calculation

Proportion calculation transforms cumulative percentages into proportions (or probabilities). This provides a clear understanding of the likelihood or frequency of data within certain intervals.

For instance, if 77.5% of fire loads are less than 600 MJ/m², then the proportion is \(\frac{77.5}{100} = 0.775\). This means approximately 77.5% of rooms fall within that range.

To calculate a proportion:

• Convert the percentage to a decimal by dividing by 100.
• For specific ranges, subtract relevant cumulative percentages.

For example, the proportion of fire loads between 600 and 1200 MJ/m² is \(\frac{98.6 - 77.5}{100} = 0.211\).

By using proportions, you can compare how data distributes over different ranges and easily identify changes in distribution.

Data Visualization

Data visualization involves techniques to graphically represent data, making it easier to understand complex ideas. A well-crafted visual can provide insights at a glance.

Visualizations like histograms help contextualize data, allowing observers to notice trends and patterns that may not be obvious from raw data alone. They reveal distributions, frequencies, and deviations.

Common types of data visualization include:

• Histograms: Show frequency distributions.
• Line graphs: Highlight trends over time.
• Pie charts: Indicate proportions within a whole.

Data visualization simplifies decision making. By transforming numerical data into visual formats, you make it more accessible to broader audiences, who may find it easier to process visual information than complex tables or raw data.