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Homework Solutions for ISyE 6739 - Summer 2009 (Modules 27-33), Assignments of Data Analysis & Statistical Methods

Georgia Institute of Technology - Main CampusData Analysis & Statistical Methods

Solutions to homework problems for isye 6739, a course offered during the summer 2009 semester. The problems cover various topics in statistics and probability, including constructing histograms, minimizing quadratic functions, normal distributions, chi-square, t-distributions, f-distributions, mean squared error, maximum likelihood estimation, and confidence intervals.

Typology: Assignments

Pre 2010

Uploaded on 08/05/2009

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ISyE 6739 — Summer 2009
Homework #7 (Covers Modules 27–33) — Due Tuesday 7/7/09
All of the following problems are from Hines, et al.
8–1. Elementary data analysis. The shelf life of a high-speed photographic film is being
investigated by the manufacturer. The following data are available (in days).
126 129 134 141
131 132 136 145
116 128 130 162
125 126 134 129
134 127 120 127
120 122 129 133
125 111 147 129
150 148 126 140
130 120 117 131
149 117 143 133
Construct a histogram and comment on the properties of the data.
8–25. Interesting algebra question. Consider the quantity Pn
i=1(xi−a)2. For what value
of ais this quantity minimized?
9–5. Normal distribution. A population of power supplies for a personal computer has
an output voltage that is normally distributed with a mean of 5.00 V and a standard
deviation of 0.10 V. A random sample of eight power supplies is selected. Specify the
sampling distribution of ¯
X.
9–23(a). χ2quantile. Find χ2
0.95,8.
9–24(a). tquantile. Find t0.25,10 .
9–25(a). Fquantile. Find F0.25,4,9.
10–1. MSE. Suppose we have a random sample of size 2nfrom a population denoted
X, and E[X] = µand Var(X) = σ2. Let
¯
X1=1
2n
2n
X
i=1
Xiand ¯
X2=1
n
n
X
i=1
Xi
pf2

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ISyE 6739 — Summer 2009

Homework #7 (Covers Modules 27–33) — Due Tuesday 7/7/

All of the following problems are from Hines, et al.

8–1. Elementary data analysis. The shelf life of a high-speed photographic film is being investigated by the manufacturer. The following data are available (in days).

Construct a histogram and comment on the properties of the data.

8–25. Interesting algebra question. Consider the quantity

∑n i=1(xi^ −^ a)

(^2). For what value

of a is this quantity minimized?

9–5. Normal distribution. A population of power supplies for a personal computer has an output voltage that is normally distributed with a mean of 5.00 V and a standard deviation of 0.10 V. A random sample of eight power supplies is selected. Specify the sampling distribution of X¯.

9–23(a). χ^2 quantile. Find χ^20. 95 , 8.

9–24(a). t quantile. Find t 0. 25 , 10.

9–25(a). F quantile. Find F 0. 25 , 4 , 9.

10–1. MSE. Suppose we have a random sample of size 2n from a population denoted X, and E[X] = μ and Var(X) = σ^2. Let

X¯ 1 = 1

2 n

∑^2 n

i=

Xi and X¯ 2 =

n

∑^ n

i=

Xi

be two estimators of μ. Which is the better estimator of μ? Explain your choice.

10–13. Geometric MLE. Let X be a geometric random variable with parameter p. Find the maximum likelihood estimator of p, based on a sample of size n.

10–41(a). Confidence interval (known variance). A civil engineer is analyzing the compressive strength of concrete. Compressive strength is approximately normally dis- tributed with a variance of σ^2 = 1000 (psi)^2. A random sample of 12 specimens has a mean compressive strength of ¯x = 3250 psi. Construct a 95% two-sided confidence interval on mean compressive strength.