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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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Most popular questions from this chapter

One piece of PVC pipe is to be inserted inside another piece. The length of the first piece is normally distributed with mean value \({\rm{20}}\) in. and standard deviation \({\rm{.5}}\) in. The length of the second piece is a normal \({\rm{rv}}\)with mean and standard deviation \({\rm{15}}\) in. and \({\rm{.4}}\) in., respectively. The amount of overlap is normally distributed with mean value \({\rm{1}}\) in. and standard deviation \({\rm{.1}}\) in. Assuming that the lengths and amount of overlap are independent of one another, what is the probability that the total length after insertion is between \({\rm{34}}{\rm{.5 }}\) in. and \({\rm{35}}\) in.?

The three measures of center introduced in this chapter are the mean, median, and trimmed mean. Two additional measures of center that are occasionally used are the midrange,which is the average of the smallest and largest observations, and the midfourth,which is the average of the two fourths. Which of these five measures of center are resistant to the effects of outliers and which are not? Explain your reasoning.

a. If a constant cis added to each \({x_i}\)in a sample, yielding \({y_i} = {x_i} + c\), how do the sample mean and median of the \({y_i}'s\)relate to the mean and median of the\({x_i}'s\)? Verify your conjectures.

b. If each \({x_i}\)is multiplied by a constant c,yielding \({y_i} = c{x_i}\), answer the question of part (a). Again, verify your conjectures.

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�<2.4 11

2.4鈭�< 2.6 13 2.6鈭�<2.8 30

2.8鈭�< 3.0 46 3.0鈭�< 3.2 66

3.2鈭�<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.

Consider the following sample of observations on coating thickness for low-viscosity paint ("Achieving a Target Value for a Manufacturing Process: \({\rm{A}}\)Case Study, \({\rm{97}}\)J. of Quality Technology, \({\rm{1992:22 - 26}}\)):

\(\begin{array}{*{20}{r}}{{\rm{.83}}}&{{\rm{.88}}}&{{\rm{.88}}}&{{\rm{1}}{\rm{.04}}}&{{\rm{1}}{\rm{.09}}}&{{\rm{1}}{\rm{.12}}}&{{\rm{1}}{\rm{.29}}}&{{\rm{1}}{\rm{.31}}}\\{{\rm{1}}{\rm{.48}}}&{{\rm{1}}{\rm{.49}}}&{{\rm{1}}{\rm{.59}}}&{{\rm{1}}{\rm{.62}}}&{{\rm{1}}{\rm{.65}}}&{{\rm{1}}{\rm{.71}}}&{{\rm{1}}{\rm{.76}}}&{{\rm{1}}{\rm{.83}}}\end{array}\)

Assume that the distribution of counting thickness is normal (a normal probability plot strongly supports this assumption).

a. Calculate a point estimate of the mean value of coating thickness, and state which estimator you used.

b. Calculate a point estimate of the median of the coating thickness distribution, and state which estimator you used.

C. Calculate a point estimate of the value that separates the largest of\({\rm{10\% }}\) all values in the thickness distribution from the remaining \({\rm{90\% }}\)and state which estimator you used. (Hint Express what you are trying to estimate in terms of \({\rm{\mu }}\)and \({\rm{\sigma }}\)

d. Estimate \({\rm{P}}\left( {{\rm{X < 1}}{\rm{.5}}} \right){\rm{,}}\)i.e., the proportion of all thickness values less than 1.5. (Hint: If you knew the values of \({\rm{\mu }}\)and \({\rm{\sigma }}\), you could calculate this probability. These values are not available, but they can be estimated.)

e. What is the estimated standard error of the estimator that you used in part (b)?
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