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Fire load (MJ/m\(^2\)) is the heat energy that could be released per square meter of floor area by combustion of contents and the structure itself. The following cumulative percentages are for fire loads in a sample of 388 rooms:

| Value (MJ/m\(^2\)) | 0 | 150 | 300 | 450 | 600 | 750 | 900 |
|--------------------|---|-----|-----|-----|-----|-----|-----|
| Cumulative % | 0 | 19.0| 37.0| 61.9| 76.5| 86.3| 93.0|

| Value (MJ/m\(^2\)) | 1050 | 1200 | 1350 | 1500 | 1650 | 1800 | 1950 |
|--------------------|------|------|------|------|------|------|------|
| Cumulative % | 95.0 | 97.8 | 98.1 | 99.5 | 99.6 | 99.8 | 100.0|

(a) What proportion of fire loads are less than 600? At least 1200? (Round your answers to three decimal places.)

(b) What proportion of the loads are between 600 and 1200? (Round your answer to three decimal places.)

Answer :

Answer:

a)

The proportion of fire loads are less than 600 is 0.765

The proportion of fire loads at least 1200 is 0.022

b)

The proportion of the loads between 600 and 1200 is 0.213

Step-by-step explanation:

The cumulative percent is given for various values of x.

Value (x) Cumulative percent

0 0

150 0.19

300 0.37

450 0.619

600 0.765

750 0.863

900 0.93

1050 0.95

1200 0.978

1350 0.981

1500 0.995

1650 0.996

1800 0.998

1950 1

a)

The proportion of fire loads are less than 600

P(x<600)=F(600)=0.765

Because we are given the cumulative percentages and the cumulative percentage for 450 will represent the proportion of fire loads that are less than 600.

The proportion of fire loads at least 1200

P(X≥1200)=1-F(1200)=1-0.978=0.022

b)

The proportion of the loads between 600 and 1200

P(600≤X≤1200)=?

P(a≤X≤b)=F(b)-F(a)

P(600≤X≤1200)=F(1200)-F(600)=0.978-0.765=0.213

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Rewritten by : Barada

Final answer:

The proportion of fire loads less than 600 MJ is 0.765, for at least 1200 MJ is 0.022, and between 600 and 1200 MJ is 0.213.

Explanation:

The question involves the analysis of fire load data to determine proportions based on cumulative percentages.

To find the proportion of fire loads less than 600 MJ/m2, we look at the given cumulative percentage for 600 MJ/m2, which states that 76.5% of the rooms have a fire load less than this value.

Therefore, the proportion is 0.765.

To determine the proportion at least 1200 MJ/m2, we observe the cumulative percentage for 1200 MJ/m2, which is 97.8%.

To find the proportion, we subtract this percentage from 100%, resulting in 2.2% or, as a proportion, 0.022.

For the proportion of the loads between 600 and 1200 MJ/m2, we subtract the cumulative percentage at 600 MJ/m2 from the cumulative percentage at 1200 MJ/m2, giving us 97.8% - 76.5%, which equals 21.3% or 0.213 as a proportion.