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Chapter 1: Q37E (page 35)

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?

Short Answer

Expert verified

The typical value is mean of the data which is 11.46million \(k{m^2}\).

Step by step solution

01

Given information

The observations on the amount of snow cover for Eurasia during the years 1970-1979 is provided.

02

Compute the typical value

Let x represents the amount of snow cover for Eurasia.

The representative or the typical value is 10% trimmed mean.

10% of 10 data points is equivalent to 1.

The data after removing the smallest and largest observation is as follows,

7.9

9.2

10

10.7

12

12.5

14.5

14.9

The 10% trimmed mean is computed as,

\(\begin{array}{c}\bar x &=& \frac{{\sum {{x_i}} }}{n}\\ &=& \frac{{7.9 + 9.2 + 10 + ... + 14.9}}{8}\\ &=& 11.46\end{array}\)

Thus, the trimmed mean is 11.46.

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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.

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The accompanying data set consists of observations,on shower-flow rate (L/min) for a sample of n=129,houses in Perth, Australia (鈥淎n Application of Bayes,Methodology to the Analysis of Diary Records in a

Water Use Study,鈥� J. Amer. Stat. Assoc., 1987: 705鈥�711):

4.6 12.3 7.1 7.0 4.0 9.2 6.7 6.9 11.5 5.1

11.2 10.5 14.3 8.0 8.8 6.4 5.1 5.6 9.6 7.5

7.5 6.2 5.8 2.3 3.4 10.4 9.8 6.6 3.7 6.4

8.3 6.5 7.6 9.3 9.2 7.3 5.0 6.3 13.8 6.2

5.4 4.8 7.5 6.0 6.9 10.8 7.5 6.6 5.0 3.3

7.6 3.9 11.9 2.2 15.0 7.2 6.1 15.3 18.9 7.2

5.4 5.5 4.3 9.0 12.7 11.3 7.4 5.0 3.5 8.2

8.4 7.3 10.3 11.9 6.0 5.6 9.5 9.3 10.4 9.7

5.1 6.7 10.2 6.2 8.4 7.0 4.8 5.6 10.5 14.6

10.8 15.5 7.5 6.4 3.4 5.5 6.6 5.9 15.0 9.6

7.8 7.0 6.9 4.1 3.6 11.9 3.7 5.7 6.8 11.3

9.3 9.6 10.4 9.3 6.9 9.8 9.1 10.6 4.5 6.2

8.3 3.2 4.9 5.0 6.0 8.2 6.3 3.8 6.0

  1. Construct a stem-and-leaf display of the data.
  2. What is a typical, or representative, flow rate?
  3. Does the display appear to be highly concentrated or spread out?
  4. Does the distribution of values appear to be reasonably symmetric? If not, how would you describe the departure from symmetry?
  5. Would you describe any observation as being far from the rest of the data (an outlier)?

Exercise 34 presented the following data on endotoxin concentration in settled dust both for a sample of urban homes and for a sample of farm homes:

U: 6.0 5.0 11.0 33.0 4.0 5.0 80.0 18.0 35.0 17.0 23.0

F: 4.0 14.0 11.0 9.0 9.0 8.0 4.0 20.0 5.0 8.9 21.0

9.2 3.0 2.0 0.3

a. Determine the value of the sample standard deviation for each sample, interpret these values, and then contrast variability in the two samples. (Hint:\(\sum {{x_i}} \)= 237.0 for the urban sample and =128.4 for the farm sample, and\(\sum {x_i^2} \)= 10,079 for the urban sample and 1617.94 for the farm sample.)

b. Compute the fourth spread for each sample and compare. Do the fourth spreads convey the same message about variability that the standard deviations do? Explain.

c. Construct a comparative boxplot (as did the cited paper)and compare and contrast the four samples.

Give one possible sample of size 4 from each of the following

populations:

a. All daily newspapers published in the United States

b. All companies listed on the New York Stock Exchange

c. All students at your college or university

d. All grade point averages of students at your college or university

Consider a sample \({x_1},{x_2},...,{x_n}\) and suppose that the values of \(\bar x\),\({s^2}\), and shave been calculated.

a. Let\({y_i} = {x_i} - \bar x\)for i=1,鈥�, n. How do the values of \({s^2}\)and sfor the\({y_i}'s\)compare to the corresponding values for the\({x_i}'s\)? Explain.

b. Let\({z_i} = \left( {{x_i} - \bar x} \right)/s\) for i=1,鈥�, n. What are the values of the sample variance and sample standard deviation for the \({z_i}'s\)?
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