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Magazine Chapter 1: Q32E (page 29)
Fire load (MJ/m2) is the heat energy that could bereleased per square meter of floor area by combustionof contents and the structure itself. The article 鈥淔ireLoads in Office Buildings鈥� (J. of Structural Engr.,
1997: 365鈥�368) gave the following cumulative percentages(read from a graph) for fire loads in a sample of388 rooms:
Value0 150 300 450 600
Cumulative %0 19.3 37.6 62.7 77.5
Value750 900 1050 1200 1350
Cumulative %87.2 93.8 95.7 98.6 99.1
Value1500 1650 1800 1950
Cumulative %99.5 99.6 99.8 100.0
a. Construct a relative frequency histogram and commenton interesting features.
b. What proportion of fire loads are less than 600? At least 1200?
c. What proportion of the loads are between 600 and1200?
Short Answer
a.
b.
The proportion of fire loads are less than 600 is 0.627.
The proportion of fire loads that are at least 1200 is 0.043.
c.
The proportion of fire loads are between 600 and 1200 is 0.182.
Step by step solution
Given information
The cumulative percentages for different values are provided.
Construct a histogram and comment
a.
The cumulative percentage data is provided.
The table for relative frequency is represented as,
Value | Cumulative % | Relative percentage | Relative frequency |
0 | 0 | 0 | 0 |
150 | 19.3 | 19.3 | 0.193 |
300 | 37.6 | 18.3 | 0.183 |
450 | 62.7 | 25.1 | 0.251 |
600 | 77.5 | 14.8 | 0.148 |
750 | 87.2 | 9.7 | 0.097 |
900 | 93.8 | 6.6 | 0.066 |
1050 | 95.7 | 1.9 | 0.019 |
1200 | 98.6 | 2.9 | 0.029 |
1350 | 99.1 | 0.5 | 0.005 |
1500 | 99.5 | 0.4 | 0.004 |
1650 | 99.6 | 0.1 | 0.001 |
1800 | 99.8 | 0.2 | 0.002 |
1950 | 100 | 0.2 | 0.002 |
Steps to construct a histogram are,
1) Determine the frequency or the relative frequency.
2) Mark the class boundaries on the horizontal axis.
3) Draw a rectangle on the horizontal axis corresponding to the frequency or relative frequency.
The histogram is represented as,
The features of the above-represented histogram are,
1)The histogram is unimodal. The value of mode is 450.
2)There are two outliers; 1800 and 1950.
3)The distribution is positively skewed.
Given information
The cumulative percentages for different values are provided.
Compute the proportion
Referring to the relative frequencies computed in part a,
b.
Let x represents the fire loads.
The proportion of fire loads are less than 600 is computed as,
\(\begin{aligned}P\left( {x < 600} \right) &= P\left( {x = 0} \right) + P\left( {x = 150} \right) + ... + P\left( {x = 450} \right)\\ &= 0 + 0.193 + ... + 0.251\\ &= 0.627\end{aligned}\)
Therefore, the proportion of fire loads are less than 600 is 0.627.
The proportion of fire loads that are at least 1200 is computed as,
\(\begin{aligned}P\left( {x \ge 1200} \right) &= P\left( {x = 1200} \right) + P\left( {x = 1350} \right) + ... + P\left( {x = 1950} \right)\\ &= 0.029 + 0.005 + ... + 0.002\\ &= 0.043\end{aligned}\)
Therefore, the proportion of fire loads that are at least 1200 is 0.043.
Given information
The cumulative percentages for different values are provided.
Compute the proportion
Let x represents the fire loads.
The proportion of fire loads are between 600and 1200 is computed as,
\(\begin{aligned}P\left( {600 < x < 1200} \right) &= P\left( {x = 750} \right) + P\left( {x = 900} \right) + P\left( {x = 1050} \right)\\ &= 0.097 + 0.066 + 0.019\\ &= 0.182\end{aligned}\)
Therefore, the proportion of fire loads are between 600and 1200 is 0.182.
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