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Allowable mechanical properties for structural design of metallic aerospace vehicles requires an approved method for statistically analyzing empirical test data. The article 鈥淓stablishing Mechanical Property Allowables for Metals鈥� (J. of Testing and Evaluation, 1998: 293鈥�299) used the accompanying data on tensile ultimate strength (ksi) as a basis for addressing the difficulties in developing such a method.

122.2 124.2 124.3 125.6 126.3 126.5 126.5 127.2 127.3

127.5 127.9 128.6 128.8 129.0 129.2 129.4 129.6 130.2

130.4 130.8 131.3 131.4 131.4 131.5 131.6 131.6 131.8

131.8 132.3 132.4 132.4 132.5 132.5 132.5 132.5 132.6

132.7 132.9 133.0 133.1 133.1 133.1 133.1 133.2 133.2

133.2 133.3 133.3 133.5 133.5 133.5 133.8 133.9 134.0

134.0 134.0 134.0 134.1 134.2 134.3 134.4 134.4 134.6

134.7 134.7 134.7 134.8 134.8 134.8 134.9 134.9 135.2

135.2 135.2 135.3 135.3 135.4 135.5 135.5 135.6 135.6

135.7 135.8 135.8 135.8 135.8 135.8 135.9 135.9 135.9

135.9 136.0 136.0 136.1 136.2 136.2 136.3 136.4 136.4

136.6 136.8 136.9 136.9 137.0 137.1 137.2 137.6 137.6

137.8 137.8 137.8 137.9 137.9 138.2 138.2 138.3 138.3

138.4 138.4 138.4 138.5 138.5 138.6 138.7 138.7 139.0

139.1 139.5 139.6 139.8 139.8 140.0 140.0 140.7 140.7

140.9 140.9 141.2 141.4 141.5 141.6 142.9 143.4 143.5

143.6 143.8 143.8 143.9 144.1 144.5 144.5 147.7 147.7

a. Construct a stem-and-leaf display of the data by first deleting (truncating) the tenths digit and then repeating each stem value five times (once for leaves 1 and 2, a second time for leaves 3 and 4, etc.). Why is it relatively easy to identify a representative strength value?

b. Construct a histogram using equal-width classes with the first class having a lower limit of 122 and an upper limit of 124. Then comment on any interesting features of the histogram.

Step by step solution

01

Given information

The data on tensile ultimate strength (ksi) as a basis for addressing the difficulties in developing is provided.

02

Construct a stem and leaf diagram and comment

a. A stem-and-leaf display provides a visual representation of the dataset.

In this case, the values are first changed by truncating the decimal number and hence obtaining the three-digit number values.

The steps to construct a stem-and-leaf for the obtained values are as follows,

1) Select the leading digit for the stem(two values- at hundreds and tens place) and trailing digits for the leaves (at one place).

2) Represent the stem digits vertically and similarly the trailing digits corresponding to the stem digits.

3) follow the rule for describing the leaves 鈥� 鈥渆ach stem value five times (once for leaves 1 and 2, a second time for leaves 3 and 4鈥�

3) Mention the units for the display.

The stem and leaf display for the provided scenario is,

12	2
12	445
12	6667777
12	889999
13	00011111111
13	222222222233333333333333
13	44444444444444444455555555555555555555
13	6666666666667777777777
13	888888888888999999
14	0000001111
14	2333333
14	444
14	77

Unit: 12|2=122

From the above display, it is easy to identify a representative strength value as the data is symmetric.

03

Construct a histogram and comment on the features

b.

The first class has a lower limit of 122.

The first-classhas anupper limit of 124.

Thus, the classes are obtained as 122-124,124-126,126-128,鈥�,146-148.

By counting the observations under each class, the frequency distribution table is obtained as,

Class	Frequency
122-124	1
124-126	3
126-128	7
128-130	6
130-132	11
132-134	25
134-136	38
136-138	22
138-140	18
140-142	10
142-144	7
144-146	3
146-148	2

The 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,

It can be observed from the above histogram that the distribution is approximately bell-shaped.

The centre lies in the interval 134-136 which can be estimated as 135.

The dispersion is not insignificant and there are no outliers present.

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