Stats- Assignment 4 /Case Study: Chest Sizes of Scottish Militiamen (p.306): Answer a, b, c, d. You must calculate results by hand (though you may use any technology of your choice to verify your answers).

306 CHAPTER 6 The Normal Distribution

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FOCUSI NG ON DATA ANALYSIS

UWEC UNDERG RADUATES

Recall from Chapter 1 (see page 34) that the Focus database and c. Based on your results from part (a), which of the variables

Focus sample contain information on the undergraduate students at considered there appear to be far from normally distributed

the University of Wisconsin – Eau Claire (UWEC). Now would be a

good time for you to review the discussion about these data sets. , If your statistlcal software package WIH accommodate the.

Begin by opening the Focus sample (FocusSample) in the ent1re Focus database (Focus), open that worksheet.

statistical software package of your cho1ce. d. Obtain a histogram for each of the following variables: high,

3. Obtain a normal probability plot of the sample data for each SChOOl percentile, cumulative GPA’ age, total earned credits,

of the following variables: high school percentile, cumulative ACT Engllsh score, ACT math score, and ACT composue

GPA, age, total earned credits, ACT English score, ACT math score.

score, and ACT composite score. e. In View of the histograms that you obtalned 1n part (d), oom-

b. Based on your results from part (a), which of the variables con- ment on your answers in parts (b) and (C)’

sidered there appear to be approximately normally distributed

g. ‘i .. CASE STU DY DISCUSSION

9:

g CHEST SIZES OF SCOTTISH MILITIAMEN

On page 263, we presented a frequency distribution for data on c. Use the table on page 263 to find the percentage of mili-

chest circumference, in inches, for 5732 Scottish militiamen. As tiamen in the survey with chest circumference between 36

mentioned there, Adolphe Quetelet used a procedure for fitting a and 41 inches, inclusive. Note: As the circumferences were

normal curve to the data based on the binomial distribution. Here rounded to the nearest inch, you are actually finding the per-

you are to accomplish that task by using techniques that you stud- centage of militiamen in the survey with chest circumference

ied in this chapter. between 35.5 and 41 .5 inches.

C 1 _ f h’ f h h , (I. Use the normal curve you identified in part (b) to obtain an ap-

a. f onstruclt a re ative- lrequenbcy :mgram 3” 61C est Circum- proximation to the percentage of militiamen in the survey with

b “thlpctfpuellattztlioliishliga;:81ngSpogilelatfonnitszihlgaidlzeveiation of the chest Circumference between 35.5 and 41.5 inches. Compare

t th t t f d ‘ t

chest Circumferences are 39.85 and 2.07, respectively. Identify your answer 0 e exac percen age oun in par (C)

the normal curve that should be used for the chest circumfer-

ences.

Q BIOGRAPHY

KN CARL FRIEDRICH GAUSS: CHILD PRODIGY

Carl Friedrich Gauss was born on April 30, 1777, in Brunswick, numbers and proved the fundamental theorem of algebra, which

Germany, the only son in a poor, semiliterate peasant family; he had previously been only partially established. Shortly thereafter,

taught himself to calculate before he could talk. At the age of 3, Gauss published his theory of numbers, which is considered one

he pointed out an error in his father’s calculations of wages. In of the most brilliant achievements in mathematics.

addition to his arithmetic experimentation, he taught himself to Gauss made important discoveries in mathematics, physics,

read. At the age of 8, Gauss instantly solved the summing of all astronomy, and statistics. Two of his major contributions to statiS-

numbers from 1 to 100. His father was persuaded to allow him to tics were the development of the least-squares method and funda-

stay in school and to study after school instead of working to help mental work with the normal distribution, often called the Gaus-

support the family. sian distribution in his honor.

Impressed by Gauss’s brilliance,the Duke ofBrunswick sup- In 1807, Gauss accepted the directorship of the observa-

ported him monetarily from the ages of 14 to 30. This patronage tory at the University of Gottingen, which ended his dependence

permitted Gauss to pursue his studies exclusively. He conceived on the Duke of Brunswick. He remained there the rest of his

most of his mathematical discoveries by the time he was 17. life. In 1833, Gauss and a colleague, Wilhelm Weber, invented a

Gauss was granted a doctorate in absentia from the university at working electric telegraph, 5 years before Samuel Morse. Gauss

Helmstedt; his doctoral thesis developed the concept of complex died in Gottingen in 1855.