Psychology 406 Assignment #1

1) Hays — Chapter 1, #14, #26, #32, #34; Chapter 2, #2

2) Suppose we denote the collection of 26 small letters {a, b, c, etc.} by the letter A.

a) Define the set A by means of a property.

b) Each subset of A must contain less than members.

c) Which of the following expressions are correct? If an expression is incorrect, explain why.

{a,e,i,o,u} ⊂ A {a,e,i,o,u} ⊆ A {a,e,i,o,u} ∈ A ∅ ⊆ ∅

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.

e) Let B = {a,e,i,o,u}, C = {a,b,c,d}, D = {a,c,d,e,i,o} A ∪ B = A ∩ C ∩ D = B − A = B̄ ∪ C =

f) Using the sets B, C, and D given above, show that

i) B ∪ (C ∩ D) = (B ∪ C) ∩ (B ∪ D) ii) B ∩ (C ∪ D) = (B ∩ C) ∪ (B ∩ D) iii) \overline{B ∩ C} = B̄ ∪ C̄ iv) \overline{B ∪ C} = B̄ ∩ C̄ Also, verify the four relationships using Venn diagrams.

g) Why is the expression B ∪ C ∩ D ambiguous.

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).

a) List the members of the sample space S. What are the elementary events?

b) If all outcomes of the three flips are equally likely, what is the probability associated with any one single elementary event?

c) Let A be the set of outcomes containing exactly one head; B the outcomes containing exactly two heads.

i) P(A) =

ii) P(B) =

iii) P(A ∪ B) =

iv) P(A ∩ B) =

v) P(Ā ∩ B) =

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^{∘}. Unable to remember how to convert to Fahrenheit, he reasoned as follows: I know that water boils at 100^{∘}C and 212^{∘}F; water freezes at 0^{∘}C and 32^{∘}F. So, if I take 37% of the difference between 212^{∘} and 32^{∘} (180) and add it to 32, I should have the Fahrenheit reading:

.37(180) + 32 = 98.6 Stanley was satisfied that his child did not have a fever. Was his reasoning correct? Explain.

5) Suppose we assign to any set A the number of elements in the set. For instance,

{a,b,d} → 3

{a} → 1, etc.

Discuss the highest type of scale that such a measurement procedure might yield.