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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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Most popular questions from this chapter

A company utilizes two different machines to manufacture parts of a certain type. During a single shift, a sample of n=20 parts produced by each machine is obtained, and the value of a particular critical dimension for each part is determined. The comparative boxplot at the bottom of this page is constructed from the resulting data. Compare and contrast the two samples.

The propagation of fatigue cracks in various aircraft parts has been the subject of extensive study in recent years. The accompanying data consists of propagation lives(flight hours/\({\bf{1}}{{\bf{0}}^{\bf{4}}}\)) to reach a given crack size in fastener holes intended for use in military aircraft (鈥淪tatistical Crack Propagation in Fastener Holes Under Spectrum Loading,鈥� J. Aircraft,1983: 1028鈥�1032):

.736 .863 .865 .913 .915 .937 .983 1.007

1.011 1.064 1.109 1.132 1.140 1.153 1.253 1.394

  1. Compute and compare the values of the sample mean and median.
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The number of contaminating particles on a silicon waferprior to a certain rinsing process was determined for eachwafer in a sample of size 100, resulting in the followingfrequencies:

Number of particles 0 1 2 3 4 5 6 7

Frequency1 2 3 12 11 15 18 10

Number of particles8 9 10 11 12 13 14

Frequency12 4 5 3 1 2 1

a. What proportion of the sampled wafers had at leastone particle? At least five particles?

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individuals:

16 18 18 26 33 41 54 56 66 68 87 91 95

98 106 109 111 118 127 127 135 145 147 149 151 168

172 183 189 190 200 210 220 229 230 233 238 244 259

294 329 403

a. Construct a stem-and-leaf display based on repeating each stem value twice, and comment on interesting features.

b. Determine the values of the fourths and the fourth spread.

c. Construct a boxplot based on the five-number summary, and comment on its features.

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e. By how much could the observation 403, currently the largest, be decreased without affecting\({f_s}\)?

Poly(3-hydroxybutyrate) (PHB), a semicrystallinepolymer that is fully biodegradable and biocompatible,is obtained from renewable resources. From a sustainabilityperspective, PHB offers many attractive propertiesthough it is more expensive to produce than standardplastics. The accompanying data on melting point(掳C) for each of 12 specimens of the polymer using adifferential scanning calorimeter appeared in the article鈥淭he Melting Behaviour of Poly(3-Hydroxybutyrate)by DSC. Reproducibility Study鈥� (Polymer Testing,2013: 215鈥�220).

180.5 181.7 180.9 181.6 182.6 181.6

181.3 182.1 182.1 180.3 181.7 180.5

Compute the following:

  1. The sample range
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  4. \({{\bf{s}}^{\bf{2}}}\)using the shortcut method
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