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Homework #1. Due: Wednesday, September 1, 1999.
IE 230. Schmeiser
Textbook: D.C. Montgomery and G.C. Runger, Applied Statistics and Probability for Engineers, John Wiley &
Sons, New York, 1999. Chapters 1 and 2.
1. (Problems 1 and 2 are designed to be worked manually. Many calculators have automatic computation of the
sample mean and sample standard deviation; for now, avoid using those functions. If you have access to such
functions, though, you might want to check your answers using them.)
(From Problem 1-1.) A sample, consisting of eight measurements of inside diameter of forged piston rings
used in an automobile engine, was taken. The data (in millimeters, mm), in the order collected, are 74.001,
74.003, 74.015, 74.000, 74.005, 74.002, 74.005, 74.004.
(a) Denote these data by x1, x2, . . . , x8, where the subscript indicates the order collected. What is the
value of x3?
(b) Compute the sample mean,

x  in1 x i / n,
where here n = 8. Include the units of measurement.
(c) Compute the sample variance

s 2  in1 ( x i  x) /(n  1).
Include the units of measurement.
(d) Re-compute the sample variance s2 using the algebraically equivalent formula

2
s 2  in1 xi2  n x ) /(n  1).
Include the units of measurement.
(e) Explain algebraically why the formula in Part (d) is equivalent to Equation (1-4) in the textbook.
(f) Explain why the formula in Part (d) is, in the textbook, called the "shortcut" method for computing s2.
(g) Compute the sample standard deviation. Include the units of measurement.
(h) When data are sorted from smallest to largest, they are called "order statistics." The usual notation is to
let x(i) denote the ith largest value. Then x(1) denotes the minimum value in the sample and x(n) denotes the
largest value. What is the value of x(3) ? Include the units of measurement.
(i) Compute the sample range r = x (n) - x (1). Include the units of measurement. Explain why this
formula is equivalent to Equation (1-6) in the textbook.
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(j) Sketch a dot diagram of these data. On the sketch, show the position of x and r.
(k) Typically the standard deviation is one-sixth to one-quarter of the range. What is the ratio in this
example?
(l) The sample range is simpler to compute than the sample standard deviation. The sample standard
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deviation is more-commonly used, however, because it "contains more information" about the data. Explain.
2. (From Problem 1-18.) Define the coded data yi = a + bxi for i = 1, 2,..., n, where a and b are constants. Coding
sometimes simplifies computations. The constant a changes the location of the data on the axis; the constant b
changes the scaling of the data.
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(a) For Problem 1 above, now denote the sample mean, sample standard deviation, and sample range by x, s X, and rX,
respectively. Let's code the original data so that the coded data are the errors from the nominal (74mm) inside
diameter, now measured in inches; that is, let b = 0.03937 inches per mm and let a = (-4mm)(0.03937)(in/mm)
= -2.91338in and b = -2.91338 in. Compute the coded data y1 , . . . , y8.
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(b) Compute the third order statistic y (3), sample mean y, sample standard deviation sY, and sample range rY. Include
the units.
(c) What is the relationship here between x(3) and y (3)? What is the general relationship in terms of a and b?
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(d) What is the relationship here between the sample means x and y? What is the general relationship in terms of a and?
(e) What is the relationship here between the sample standard deviations s X and sY. What is the general relationship in
terms of a and b?
(f) What is the relationship here between the sample ranges rX and rY. What is the general relationship in terms of a and
b?
3. (Do the following analysis using a computer spread sheet. E-mail the spread sheet to "[email protected]". In the
subject line, say "Prob 1-6, your.name". The reply will be electronic, so mail from the account where you want
your reply.)
(From Problem 1-6.) The following data are direct solar-intensity measurements (in watts/meter2) on different days at a
location in southern Spain: 562, 869, 708, 775, 775, 704, 809, 856, 655, 806, 878, 909, 918, 558, 768, 870, 918, 940,
946, 661, 820, 898, 935, 952, 957, 693, 835, 905, 939, 955, 960, 498, 653, 730, 753.
(a) Enter the data into a column. Leave some blank rows above the data and at least one column to the left. (b) In the
first row or two or three, provide some heading information, including your name, date, problem number, and a brief
description of the meaning of the data.
(c) In a column to the left, enter the integers 1 through n, where n is the sample size.
(d) Below the heading and above the data, compute the sample mean. (In MSExcel, you may use "average".) In the cell
to its left, label it.
(e) Below the sample mean, compute the sample standard deviation. (In MSExcel, you may use "stdev".) In the cell to
its left, label it.
(f) In a new, third, column, create the squared values x2i for i = 1 , . . . , n.
(g) Above the new column of squared data values, and in the same row as the sample standard deviation of Part (d),
compute the standard deviation via the "shortcut" method, which uses only this new column of squared data
values, the sample-size cell, and the sample-mean cell. (In MSExcel, you may use "sum".)
(h) In a fourth column, place the values i /(n +1) for i = 1, 2,..., n.
(i) Sort the original data and place the order statistics x (1), . . . , x (n) into a fifth column. Highlight in red the sample
median. Highlight in green the first and third sample quartiles. Highlight in blue x(1) and x(n)
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(j) Above this fifth column, compute the sample range. In the cell to its left, label it.
(k) Create a scatter plot from the fourth and fifth columns to obtain an empirical cumulative distribution plot (similar to
Figure 2-7 in the textbook). Label the plot and both axes.
Quiz material for Monday, August 30, 1999.
1. English alphabet. Be able to write each of the 26 letters in both upper and lower case in a unique manner. In
particular, there should be no ambiguity about the case of a letter. (See class handout.)
2. Greek alphabet. Given a letter's name, such as "lambda", be able to write it both in upper and lowercase. (See class
handout.)
3. Probability versus statistics. Understand the difference. For example, given a situation, be able to identify whether it
is an application of probability or statistics.
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