Chapter 7.7, Problem 92E

a.

To determine

Construct the histogram for body mass of the 50 difference bird species based on the class intervals.

Identify whether transformation of the data is desirable or not.

Answer to Problem 92E

The histogram for body mass of the 50 difference bird species based on the class intervals is given below:

The transformation of the data is desirable.

Explanation of Solution

Calculation:

It is given that the body mass of the 50 different bird species.

Software procedure:

Step-by-step procedure to construct the histogram using the MINITAB software:

• Choose Graph > Histogram.
• Select Simple, and then click OK.
• In Graph variables, enter the corresponding column of Body mass.
• Click OK.
• Double click on x axis and select Binning.
• Choose Cutpoint and enter Number of intervals as 9.
• Enter 5 10 15 20 25 30 40 50 100 500 under Midpoint/Cutpoint positions.
• Click Ok.

From the histogram, it can be observed that most of the data values fall towards the left of the distribution extending the tail towards the right indicating that the distribution of body mass is positively skewed. Thus, for making the data symmetric a transformation is needed.

Hence, the transformation of the data is desirable.

b.

To determine

Construct the histogram by transforming the data using logarithms.

Identify whether the log transformation was successful in producing a more symmetric distribution or not.

Answer to Problem 92E

The histogram by transforming the data using logarithms is given below:

Yes, the log transformation was successful in producing a more symmetric distribution.

Explanation of Solution

Calculation:

It is given that the body mass of the 50 different bird species. The log for the data values is obtained as shown below:

Body Mass	Log (Body Mass)	Transformed values
7.7	log (7.7)	0.88649
10.1	log (10.1)	1.00432
21.6	log (21.6)	1.33445
8.6	log (8.6)	0.93450
12	log (12)	1.07918
11.4	log (11.4)	1.05690
16.6	log (16.6)	1.22011
9.4	log (9.4)	0.97313
11.5	log (11.5)	1.06070
9	log (9)	0.95424
8.2	log (8.2)	0.91381
20.2	log (20.2)	1.30535
48.5	log (48.5)	1.68574
21.6	log (21.6)	1.33445
26.1	log (26.1)	1.41664
6.2	log (6.2)	0.79239
19.1	log (19.1)	1.28103
21	log (21)	1.32222
28.1	log (28.1)	1.44871
10.6	log (10.6)	1.02531
31.6	log (31.6)	1.49969
6.7	log (6.7)	0.82607
5	log (5)	0.69897
68.8	log (68.8)	1.83759
23.9	log (23.9)	1.37840
19.8	log (19.8)	1.29667
20.1	log (20.1)	1.30320
6	log (6)	0.77815
99.6	log (99.6)	1.99826
19.8	log (19.8)	1.29667
16.5	log (16.5)	1.21748
9	log (9)	0.95424
448	log (448)	2.65128
21.3	log (21.3)	1.32838
17.4	log (17.4)	1.24055
36.9	log (36.9)	1.56703
34	log (34)	1.53148
41	log (41)	1.61278
15.9	log (15.9)	1.20140
12.5	log (12.5)	1.09691
10.2	log (10.2)	1.00860
31	log (31)	1.49136
21.5	log (21.5)	1.33244
11.9	log (11.9)	1.07555
32.5	log (32.5)	1.51188
9.8	log (9.8)	0.99123
93.9	log (93.9)	0.97267
10.9	log (10.9)	1.03743
19.6	log (19.6)	1.29226
14.5	log (14.5)	1.16137

Software procedure:

Step-by-step procedure to obtain histogram using MINITAB:

• Choose Graph > Histogram.
• Choose Simple, and then click OK.
• In Graph variables, enter the corresponding column of Logarithm of Body mass.
• Click OK.
• On graph right click on X axis, select edit X scale.
• Select Binning.
• Under Interval Type, Choose cut point.
• Click OK.

Justification: It can be observed that the histogram of the transformed data (using logarithms) is more near to symmetrical shape when compared to the untransformed data. This shows that using the transformation of logarithms produce a more symmetrical distribution for body mass.

Hence, the log transformation was successful in producing a more symmetric distribution.

c.

To determine

Construct the histogram by transforming the data using $\frac{1}{\sqrt{\text{original value}}}$.

Identify whether the transformation $\frac{1}{\sqrt{\text{original value}}}$ appears to resemble a normal distribution or not.

Answer to Problem 92E

The histogram by transforming the data using $\frac{1}{\sqrt{\text{original value}}}$ is given below:

Yes, the transformation $\frac{1}{\sqrt{\text{original value}}}$ appears to resemble approximately a normal distribution.

Explanation of Solution

Calculation:

The transformation for the data values using $\frac{1}{\sqrt{\text{original value}}}$ is obtained as shown below:

Body Mass	$\frac{1}{\sqrt{\begin{array}{l}\text{original value }\\ \text{of body mass}\end{array}}}$	Transformed values
7.7	$\frac{1}{\sqrt{7.7}}$	0.360375
10.1	$\frac{1}{\sqrt{10.1}}$	0.314658
21.6	$\frac{1}{\sqrt{21.6}}$	0.215166
8.6	$\frac{1}{\sqrt{8.6}}$	0.340997
12	$\frac{1}{\sqrt{12}}$	0.288675
11.4	$\frac{1}{\sqrt{11.4}}$	0.296174
16.6	$\frac{1}{\sqrt{16.6}}$	0.245440
9.4	$\frac{1}{\sqrt{9.4}}$	0.326164
11.5	$\frac{1}{\sqrt{11.5}}$	0.294884
9	$\frac{1}{\sqrt{9}}$	0.333333
8.2	$\frac{1}{\sqrt{8.2}}$	0.349215
20.2	$\frac{1}{\sqrt{20.2}}$	0.222497
48.5	$\frac{1}{\sqrt{48.5}}$	0.143592
21.6	$\frac{1}{\sqrt{21.6}}$	0.215166
26.1	$\frac{1}{\sqrt{26.1}}$	0.195740
6.2	$\frac{1}{\sqrt{6.2}}$	0.401610
19.1	$\frac{1}{\sqrt{19.1}}$	0.228814
21	$\frac{1}{\sqrt{21}}$	0.218218
28.1	$\frac{1}{\sqrt{28.1}}$	0.188646
10.6	$\frac{1}{\sqrt{10.6}}$	0.307148
31.6	$\frac{1}{\sqrt{31.6}}$	0.177892
6.7	$\frac{1}{\sqrt{6.7}}$	0.386334
5	$\frac{1}{\sqrt{5}}$	0.447214
68.8	$\frac{1}{\sqrt{68.8}}$	0.120561
23.9	$\frac{1}{\sqrt{23.9}}$	0.204551
19.8	$\frac{1}{\sqrt{19.8}}$	0.224733
20.1	$\frac{1}{\sqrt{20.1}}$	0.223050
6	$\frac{1}{\sqrt{6}}$	0.408248
99.6	$\frac{1}{\sqrt{99.6}}$	0.100201
19.8	$\frac{1}{\sqrt{19.8}}$	0.224733
16.5	$\frac{1}{\sqrt{16.5}}$	0.246183
9	$\frac{1}{\sqrt{9}}$	0.333333
448	$\frac{1}{\sqrt{448}}$	0.017546
21.3	$\frac{1}{\sqrt{21.3}}$	0.216676
17.4	$\frac{1}{\sqrt{17.4}}$	0.239732
36.9	$\frac{1}{\sqrt{36.9}}$	0.164622
34	$\frac{1}{\sqrt{34}}$	0.171499
41	$\frac{1}{\sqrt{41}}$	0.156174
15.9	$\frac{1}{\sqrt{15.9}}$	0.250785
12.5	$\frac{1}{\sqrt{12.5}}$	0.282843
10.2	$\frac{1}{\sqrt{10.2}}$	0.313112
31	$\frac{1}{\sqrt{31}}$	0.179605
21.5	$\frac{1}{\sqrt{21.5}}$	0.215666
11.9	$\frac{1}{\sqrt{11.9}}$	0.289886
32.5	$\frac{1}{\sqrt{32.5}}$	0.175412
9.8	$\frac{1}{\sqrt{9.8}}$	0.319438
93.9	$\frac{1}{\sqrt{93.9}}$	0.103197
10.9	$\frac{1}{\sqrt{10.9}}$	0.302891
19.6	$\frac{1}{\sqrt{19.6}}$	0.225877
14.5	$\frac{1}{\sqrt{14.5}}$	0.262613

Software procedure:

Step-by-step procedure to obtain histogram using MINITAB:

• Choose Graph > Histogram.
• Choose Simple, and then click OK.
• In Graph variables, enter the corresponding column of 1/sqrt(Body mass).
• Click OK.
• On graph right click on X axis, select edit X scale.
• Select Binning.
• Under Interval Type, Choose cut point.
• Click OK.

Justification: It can be observed that the histogram of the transformed data (using $\frac{1}{\sqrt{\text{Body mass}}}$) is more near to symmetrical shape when compared to the untransformed data and even the transformation using logarithms. This shows that using the transformation $\frac{1}{\sqrt{\text{Body mass}}}$, has produced a more symmetrical distribution making it approximately normal.

Hence, the transformation $\frac{1}{\sqrt{\text{original value}}}$ appears to resemble approximately a normal distribution.