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Chapter 1: Q24E (page 27)

The accompanying data set consists of observations on shear strength (lb) of ultrasonic spot welds made on a certain type of alclad sheet. Construct a relative frequency histogram based on ten equal-width classes with boundaries 4000, 4200, 鈥�. [The histogram will agree with the one in 鈥淐omparison of Properties of Joints Prepared by Ultrasonic Welding and Other Means鈥� (J. of Aircraft, 1983: 552鈥�556).] Comment on its features.

5434

4948

4521

4570

4990

5702

5241

5112

5015

4659

4806

4637

5670

4381

4820

5043

4886

4599

5288

5299

4848

5378

5260

5055

5828

5218

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5027

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4772

5133

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4618

4848

5089

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5333

5164

5342

5069

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4925

5001

4803

4951

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5256

5207

5621

4918

5138

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4500

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4173

5296

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5170

4740

5173

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4900

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5245

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5275

5419

5205

4452

5227

5555

5388

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4681

5076

4774

4931

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5582

4308

4823

4417

5364

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5188

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5042

5189

4986

Short Answer

Expert verified

The histogram is represented as,

Step by step solution

01

Given information

The data for the shear strength (lb) of ultrasonic spot welds made on a certain type of alclad sheet is provided.

02

Construct a histogram and state its features

Let x represents the shear strength (lb) of ultrasonic spot welds.

The total number of observations is 100.

The relative frequency is computed as,

\({\rm{relative frequency }} = \frac{{frequency}}{{Total\;number\;of\;observations}}\)

The table representing the relative frequency is as follows,

x

Frequency

Relative Frequency

4000-4200

1

0.01

4200-4400

2

0.02

4400-4600

9

0.09

4600-4800

12

0.12

4800-5000

19

0.19

5000-5200

22

0.22

5200-5400

20

0.2

5400-5600

7

0.07

5600-5800

7

0.07

5800-6000

1

0.01

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-diagram, it can be concluded that the distribution is approximately negatively-skewed. There are no outliers present in the data. The typical or representative value of x is 5049.

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Most popular questions from this chapter

Compute the sample median, 25% trimmed mean, 10% trimmed mean, and sample mean for the lifetime data given in Exercise 27, and compare these measures.

The cdf for \({\rm{X( = measurement error)}}\) of Exercise \({\rm{3}}\) is

\({\rm{F(x) = }}\left\{ {\begin{array}{*{20}{c}}{\rm{0}}&{{\rm{x < - 2}}}\\{\frac{{\rm{1}}}{{\rm{2}}}{\rm{ + }}\frac{{\rm{3}}}{{{\rm{32}}}}\left( {{\rm{4x - }}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right)}&{{\rm{ - 2}} \le {\rm{x < 2}}}\\{\rm{1}}&{{\rm{2}} \le {\rm{x}}}\end{array}} \right.\)

a. Compute \({\rm{P(X < 0)}}\).

b. Compute \({\rm{P( - 1 < X < 1)}}\).

c. Compute \({\rm{P(}}{\rm{.5 < X)}}\).

d. Verify that \({\rm{f(x)}}\) is as given in Exercise \({\rm{3}}\) by obtaining \({\rm{F'(x)}}\).

e. Verify that \(\widetilde {\rm{\mu }}{\rm{ = 0}}\).

The accompanying data came from a study of collusion inbidding within the construction industry (鈥淒etection ofCollusive Behavior,鈥� J. of Construction Engr. AndMgmnt, 2012: 1251鈥�1258).

No.Bidders

No.Contracts

2

7

3

20

4

26

5

16

6

11

7

9

8

6

9

8

10

3

11

2

a. What proportion of the contracts involved at mostfive bidders? At least five bidders?

b. What proportion of the contracts involved betweenfive and 10 bidders, inclusive? Strictly between fiveand 10 bidders?

c. Construct a histogram and comment on interestingfeatures.

Once an individual has been infected with a certain disease, let \({\rm{X}}\) represent the time (days) that elapses before the individual becomes infectious. The article proposes a Weibull distribution with \({\rm{\alpha = 2}}{\rm{.2}}\), \({\rm{\beta = 1}}{\rm{.1}}\), and \({\rm{\gamma = 0}}{\rm{.5}}\).

a. Calculate \({\rm{P(1 < X < 2)}}\).

b. Calculate \({\rm{P(X > 1}}{\rm{.5)}}\).

c. What is the \({\rm{90th}}\) percentile of the distribution?

d. What are the mean and standard deviation of \({\rm{X}}\)?

The article 鈥淒etermination of Most RepresentativeSubdivision鈥� (J. of Energy Engr., 1993: 43鈥�55) gavedata on various characteristics ofsubdivisions that couldbe used in deciding whether to provide electrical powerusing overhead lines or underground lines. Here are thevalues of the variable x=total length of streets within asubdivision:

1280

5320

4390

2100

1240

3060

4770

1050

360

3330

3380

340

1000

960

1320

530

3350

540

3870

1250

2400

960

1120

2120

450

2250

2320

2400

3150

5700

5220

500

1850

2460

5850

2700

2730

1670

100

5770

3150

1890

510

240

396

1419

2109

a. Construct a stem-and-leaf display using the thousandsdigit as the stem and the hundreds digit as theleaf, and comment on the various features of thedisplay.

b. Construct a histogram using class boundaries 0, 1000, 2000, 3000, 4000, 5000, and 6000. What proportion of subdivisions have a total length less than 2000? Between 2000 and 4000? How would you describe the shape of the histogram?
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