/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q28E The accompanying frequency distr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The article 鈥淪now Cover and Temperature Relationships in North America and Eurasia鈥� (J. Climate and Applied Meteorology, 1983: 460鈥�469) used statistical techniques to relate the amount of snow cover on each continent to average continental temperature. Data presented there included the following ten observations on October snow cover for Eurasia during the years 1970鈥�1979 (in million\({\bf{k}}{{\bf{m}}^{\bf{2}}}\)):

6.5 12.0 14.9 10.0 10.7 7.9 21.9 12.5 14.5 9.2

What would you report as a representative, or typical, value of October snow cover for this period, and what prompted your choice?

Fretting is a wear process that results from tangential oscillatory movements of small amplitude in machine parts. The article 鈥淕rease Effect on Fretting Wear of Mild Steel鈥� (Industrial Lubrication and Tribology, 2008: 67鈥�78) included the following data on volume wear (1024 mm3 ) for base oils having four different viscosities.

Viscosity Wear
20.4 58.8 30.8 27.3 29.9 17.7 76.5
30.2 44.5 47.1 48.7 41.6 32.8 18.3
89.4 73.3 57.1 66.0 93.8 133.2 81.1
252.6 30.6 24.2 16.6 38.9 28.7 23.6

a. The sample coefficient of variation 100s/x-bar assesses the extent of variability relative to the mean (specifically, the standard deviation as a percentage of the mean). Calculate the coefficient of variation for the
sample at each viscosity. Then compare the results and comment.
b. Construct a comparative boxplot of the data and comment on interesting features.

An experiment to study the lifetime (in hours) for acertain type of component involved putting ten components into operation and observing them for 100hours. Eight of the components failed during that

period, and those lifetimes were recorded. Denote the lifetimes of the two components still functioning after100 hours by 100+. The resulting sample observations were

48 79 100+ 35 92 86 57 100+ 17 29

Which of the measures of center discussed in this section can be calculated, and what are the values of those measures? (Note:The data from this experiment is said to be 鈥渃ensored on the right.鈥�)

A transformation of data values by means of some mathematical function, such as\(\sqrt x \)or\(1/x\), can often yield a set of numbers that has 鈥渘icer鈥� statistical properties than the original data. In particular, it may be possible to find a function for which the histogram of transformed values is more symmetric (or, even better, more like a bell-shaped curve) than the original data. As an example, the article 鈥淭ime Lapse Cinematographic Analysis of Beryllium鈥�Lung FibroblastInteractions鈥� (Environ. Research,1983: 34鈥�43) reported the results of experiments designed to study the behavior of certain individual cells that had been exposed to beryllium. An important characteristic of such an individual cell is its interdivision time (IDT). IDTs were determined for a large number of cells, both in exposed (treatment) and unexposed(control) conditions. The authors of the articleused a logarithmic transformation, that is, transformed value=log(original value). Consider the following representative IDT data:

IDT log10(IDT) IDT log10(IDT) IDT log10(IDT)

28.1 1.45 60.1 1.78 21.0 1.32

31.2 1.49 23.7 1.37 22.3 1.35

13.7 1.14 18.6 1.27 15.5 1.19

46.0 1.66 21.4 1.33 36.3 1.56

25.8 1.41 26.6 1.42 19.1 1.28

16.8 1.23 26.2 1.42 38.4 1.58

34.8 1.54 32.0 1.51 72.8 1.86

62.3 1.79 43.5 1.64 48.9 1.69

28.0 1.45 17.4 1.24 21.4 1.33

17.9 1.25 38.8 1.59 20.7 1.32

19.5 1.29 30.6 1.49 57.3 1.76

21.1 1.32 55.6 1.75 40.9 1.61

31.9 1.50 25.5 1.41

28.9 1.46 52.1 1.72

Use class intervals 10-<20, 20-<30,鈥� to construct a histogram of the original data. Use intervals1.1-<1.2, 1.2-<1.3,鈥� to do the same for the transformed data. What is the effect of the transformation?

Consider the strength data for beams given in Example

1.2.

a. Construct a stem-and-leaf display of the data. What appears to be a representative strength value? Do the observations appear to be highly

concentrated about the representative value or rather spread out?

b. Does the display appear to be reasonably symmetric about a representative value, or would you describe its shape in some other way?

c. Do there appear to be any outlying strength values?

d. What proportion of strength observations in this sample exceeds 10 MPa?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.