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Psychology 406
Assignment \#1
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1) Hays --- Chapter 1, \#14, \#26, \#32, \#34; Chapter 2, \#2
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2) Suppose we denote the collection of 26 small letters \{a, b, c,
etc.\} by the letter A.
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a) Define the set A by means of a property.
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b) Each subset of A must contain less than \rule{.5in}{.01in}
members.
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c) Which of the following expressions are correct? If an expression
is incorrect, explain why.
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\{a,e,i,o,u\} $\subset$ A
\{a,e,i,o,u\} $\subseteq$ A
\{a,e,i,o,u\} $\in$ A
$\emptyset$ $\subseteq$ $\emptyset$
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Consider A to be the universal set.
d) Define the complement of the set of vowels \{a,e,i,o,u\} by means
of a property.
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e) Let B = \{a,e,i,o,u\}, C = \{a,b,c,d\}, D = \{a,c,d,e,i,o\}
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A $\cup$ B =
A $\cap$ C $\cap$ D =
B $-$ A =
$\bar{\mathrm{B}}$ $\cup$ C =
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f) Using the sets B, C, and D given above, show that
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i) B $\cup$ (C $\cap$ D) = (B $\cup$ C) $\cap$ (B $\cup$ D)
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ii) B $\cap$ (C $\cup$ D) = (B $\cap$ C) $\cup$ (B $\cap$ D)
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iii) $\overline{\mathrm{B} \cap \mathrm{C}}$ = $\bar{\mathrm{B}}$
$\cup$ $\bar{\mathrm{C}}$
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iv) $\overline{\mathrm{B} \cup \mathrm{C}}$ = $\bar{\mathrm{B}}$
$\cap$ $\bar{\mathrm{C}}$
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Also, verify the four relationships using Venn diagrams.
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g) Why is the expression B $\cup$ C $\cap$ D ambiguous.
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3) Suppose a fair coin (probability of heads is 1/2) is flipped
three times and the outcomes recorded as a unit, e.g., (H,T,H).
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a) List the members of the sample space $\mathcal{S}$. What are the
elementary events?
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b) If all outcomes of the three flips are equally likely, what is
the probability associated with any one single elementary event?
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c) Let A be the set of outcomes containing exactly one head; B the
outcomes containing exactly two heads.
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i) P(A) =
ii) P(B) =
iii) P(A $\cup$ B) =
iv) P(A $\cap$ B) =
v) P($\bar{\mathrm{A}}$ $\cap$ B) =
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4) Stanley Student is typical of most graduate TAs, but has one
distinct disadvantage. Because his wife is a nurse, the thermometer
they have is calibrated in Centigrade rather than Fahrenheit. One
night, while his wife was working, their new baby started to cry ---
much more than usual. Stanley decided to take her temperature and
after much more crying, succeeded in getting a reading of
37$^{\circ}$. Unable to remember how to convert to Fahrenheit, he
reasoned as follows: I know that water boils at 100$^{\circ}$C and
212$^{\circ}$F; water freezes at 0$^{\circ}$C and 32$^{\circ}$F. So,
if I take 37\% of the difference between 212$^{\circ}$ and
32$^{\circ}$ (180) and add it to 32, I should have the Fahrenheit
reading:
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.37(180) + 32 = 98.6
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\noindent Stanley was satisfied that his child did not have a fever.
Was his reasoning correct? Explain.
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5) Suppose we assign to any set A the number of elements in the set.
For instance,
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\{a,b,d\} $\rightarrow$ 3
\{a\} $\rightarrow$ 1, etc.
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Discuss the highest type of scale that such a measurement procedure
might yield.
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