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Magazine Chapter 1: Q28E (page 28)
The accompanying frequency distribution on depositedenergy (mJ) was extracted from the article 鈥淓xperimentalAnalysis of Laser-Induced Spark Ignition of LeanTurbulent Premixed Flames鈥� (Combustion and Flame,2013: 1414鈥�1427).
1.0鈭�< 2.0 5 2.0鈭�&濒迟;2.4 11
2.4鈭�< 2.6 13 2.6鈭�&濒迟;2.8 30
2.8鈭�< 3.0 46 3.0鈭�< 3.2 66
3.2鈭�&濒迟;3.4 133 3.4鈭�< 3.6 141
3.6鈭�< 3.8 126 3.8鈭�< 4.0 92
4.0鈭�< 4.2 73 4.2鈭�< 4.4 38
4.4鈭�< 4.6 19 4.6鈭�< 5.0 11
a. What proportion of these ignition trials resulted in adeposited energy of less than 3 mJ?
b. What proportion of these ignition trials resulted in adeposited energy of at least 4 mJ?
c. Roughly what proportion of the trials resulted in adeposited energy of at least 3.5 mJ?
d. Construct a histogram and comment on its shape.
Short Answer
a.
The proportion of these ignition trials resulted in deposited energy of less than 3 mJis 0.131.
b.
The proportion of these ignition trials resulted in deposited energy of at least 4 mJ is 0.175.
c.
The proportion of these ignition trials resulted in deposited energy of at least 3.5 mJ is 0.447.
d. The histogram is represented as,
Step by step solution
Given information
A frequency distribution on deposited energy is provided.
Compute the proportion
The relative frequency is computed as,
\({\rm{relative frequency }} = \frac{{frequency}}{{Total\;number\;of\;observations}}\)
The relative frequency distribution is represented as,
Class | Frequency | Relative Frequency |
1.0鈭�&濒迟;2 | 5 | 0.006 |
2.0鈭�&濒迟;2.4 | 11 | 0.014 |
2.4鈭�&濒迟;2.6 | 13 | 0.016 |
2.6鈭�&濒迟;2.8 | 30 | 0.037 |
2.8鈭�&濒迟;3 | 46 | 0.057 |
3.0鈭�&濒迟;3.2 | 66 | 0.082 |
3.2鈭�&濒迟;3.4 | 133 | 0.165 |
3.4鈭�&濒迟;3.6 | 141 | 0.175 |
3.6鈭�&濒迟;3.8 | 126 | 0.157 |
3.8鈭�&濒迟;4 | 92 | 0.114 |
4.0鈭�&濒迟;4.2 | 73 | 0.091 |
4.2鈭�&濒迟;4.4 | 38 | 0.047 |
4.4鈭�&濒迟;4.6 | 19 | 0.024 |
4.6鈭�&濒迟;5 | 11 | 0.014 |
a.
Let x represents the deposited energy.
The proportion of these ignition trials resulted in deposited energy of less than 3 mJis computed as,
\(\begin{aligned}P\left( {x < 3} \right) &= P\left( {1.0 - < 2} \right) + P\left( {2.0 - < 2.4} \right) + P\left( {2.4 - < 2.6} \right) + ... + P\left( {2.8 - < 3} \right)\\ &= 0.006 + 0.014 + 0.016 + ... + 0.057\\ &= 0.131\end{aligned}\)
Thus, theproportion of these ignition trials resulted in deposited energy of less than 3 mJis 0.131.
Given information
A frequency distribution on deposited energy is provided.
Compute the proportion
The relative frequency is computed as,
\({\rm{relative frequency }} = \frac{{frequency}}{{Total\;number\;of\;observations}}\)
The relative frequency distribution is represented as,
Class | Frequency | Relative Frequency |
1.0鈭�&濒迟;2 | 5 | 0.006 |
2.0鈭�&濒迟;2.4 | 11 | 0.014 |
2.4鈭�&濒迟;2.6 | 13 | 0.016 |
2.6鈭�&濒迟;2.8 | 30 | 0.037 |
2.8鈭�&濒迟;3 | 46 | 0.057 |
3.0鈭�&濒迟;3.2 | 66 | 0.082 |
3.2鈭�&濒迟;3.4 | 133 | 0.165 |
3.4鈭�&濒迟;3.6 | 141 | 0.175 |
3.6鈭�&濒迟;3.8 | 126 | 0.157 |
3.8鈭�&濒迟;4 | 92 | 0.114 |
4.0鈭�&濒迟;4.2 | 73 | 0.091 |
4.2鈭�&濒迟;4.4 | 38 | 0.047 |
4.4鈭�&濒迟;4.6 | 19 | 0.024 |
4.6鈭�&濒迟;5 | 11 | 0.014 |
b.
Let x represents the deposited energy.
The proportion of these ignition trials resulted in deposited energy of at least 4 mJ is computed as,
\(\begin{aligned}P\left( {x \ge 4} \right) &= P\left( {4.0 - < 4.2} \right) + P\left( {4.2 - < 4.4} \right) + P\left( {4.4 - < 4.6} \right) + P\left( {4.6 - < 5} \right)\\ &= 0.091 + 0.047 + 0.024 + 0.014\\ &= 0.175\end{aligned}\)
Thus, the proportion of these ignition trials resulted in deposited energy of at least 4 mJ is 0.175.
Given information
A frequency distribution on deposited energy is provided.
Compute the proportion
The relative frequency is computed as,
\({\rm{relative frequency }} = \frac{{frequency}}{{Total\;number\;of\;observations}}\)
The relative frequency distribution is represented as,
Class | Frequency | Relative Frequency |
1.0鈭�&濒迟;2 | 5 | 0.006 |
2.0鈭�&濒迟;2.4 | 11 | 0.014 |
2.4鈭�&濒迟;2.6 | 13 | 0.016 |
2.6鈭�&濒迟;2.8 | 30 | 0.037 |
2.8鈭�&濒迟;3 | 46 | 0.057 |
3.0鈭�&濒迟;3.2 | 66 | 0.082 |
3.2鈭�&濒迟;3.4 | 133 | 0.165 |
3.4鈭�&濒迟;3.6 | 141 | 0.175 |
3.6鈭�&濒迟;3.8 | 126 | 0.157 |
3.8鈭�&濒迟;4 | 92 | 0.114 |
4.0鈭�&濒迟;4.2 | 73 | 0.091 |
4.2鈭�&濒迟;4.4 | 38 | 0.047 |
4.4鈭�&濒迟;4.6 | 19 | 0.024 |
4.6鈭�&濒迟;5 | 11 | 0.014 |
c.
Let x represents the deposited energy.
The proportion of these ignition trials resulted in deposited energy of at least 3.5 mJ is computed as,
\(\begin{aligned}P\left( {x \ge 3.5} \right) &= P\left( {3.6 - < 3.8} \right) + P\left( {3.8 - < 4} \right) + P\left( {4 - < 4.2} \right) + ... + P\left( {4.6 - < 5} \right)\\ &= 0.157 + 0.114 + 0.091 + ... + 0.014\\ &= 0.447\end{aligned}\)
Thus, the proportion of these ignition trials resulted in deposited energy of at least 3.5 mJ is 0.447.
Given information
A frequency distribution on deposited energy is provided.
Construct a histogram and comment on the shape
Referring to the relative frequencies computed in part a,
d.
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,
From the above-represented histogram, the shape of the distribution is approximately symmetric.
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