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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

Expert verified

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

01

Given information

A frequency distribution on deposited energy is provided.

02

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.

03

Given information

A frequency distribution on deposited energy is provided.

04

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.

05

Given information

A frequency distribution on deposited energy is provided.

06

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.

07

Given information

A frequency distribution on deposited energy is provided.

08

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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