Math Answers

How many ways are there to select 12 countries in the United Nations to serve on a council if 3 is selected from a block of 53, 1 are selected from a block of 62 and 8 are selected from the remaining 74 countries?

Complete { (2, 0, 3)} to form an orthonormal basis of R^3

Construct a truth table for each of these compound propositions.

(i) (p → q) ↔ (¬q → ¬p) (16 marks)

ii) p ⊕ (p ∨ q) (8 marks)

(i) Determine by using truth tables if (p ∧ q) → p is a tautology, contradiction or a contingency. Give reasons for your answer. (6 marks)

(ii) Show that ¬(p ⊕ q) and p ↔ q are logically equivalent. (6 marks)

In the Super-Mega lottery there are 50 numbers (1 to 50), a player chooses ten different numbers and hopes that these get drawn. If the player's numbers get drawn, he/she wins an obscene amount of money. The table below displays the frequency with which classes of numbers are chosen (not drawn). These numbers came from a sample of 214 chosen numbers.

Chosen Numbers (𝑛=214)

1 to 10 11 to 20. 21 to 30. 31 to 40. 41 to 50

Count. 54. 46. 37. 54. 23

Test the claim that chosen numbers are not evenly distributed across the five classes. Test this claim at the 0.10 significance level.

(a) Find the test statistic.

(b) Find the critical value.

 A student wants to see if the correct answers to multiple choice problems are evenly distributed. She heard a rumor that if you don't know the answer, you should always pick C. In a sample of 78 multiple-choice questions from prior tests and quizzes, the distribution of correct answers are given in the table below. In all of these questions, there were four options {A, B, C, D}.

Correct Answers (𝑛=78)

A B. C. D

Count. 10. 23. 10. 35

Test the claim that correct answers for all multiple-choice questions are not evenly distributed. Test this claim at the 0.10 significance level.

(a) Find the test statistic.

(b) Find the critical value.

Among drivers who have had a car crash in the last year, 180 were randomly selected and categorized by age, with the results listed in the table below.

Age. Under 25. 25-44. 45-64. Over 64

Drivers. 72 40 28 40

If all ages have the same crash rate, we would expect (because of the age distribution of licensed drivers) the given categories to have 16%, 44%, 27%, 13% of the subjects, respectively. At the 0.05 significance level, test the claim that the distribution of crashes conforms to the distribution of ages.

The test statistic is 𝜒2 =

The critical value is 𝜒2 =

The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments from a SRS of small colleges were counted, and the results are shown in the table below.

Dept. Math. Physics. Chemistry. Linguistics English

Men 58 74 36 12 25

Women. 5 7 10 11 20

Test the claim that the gender of a professor is independent of the department. Use the significance level 𝛼=0.01

(a) The test statistic is 𝜒2 =

(b) The critical value is 𝜒2 =

Let the universe of discourse be the set of all integers. Let p; q; r; s, and t be as follows: p(x):x>0,q(x):xiseven,r(x):xisaperfectsquare,s(x):xis(exactly)divisibleby4, t(x):x is (exactly) divisible by 5. (8 marks)

Write the following statements using quantifiers and logical connectives

i. At least one integer is even.

ii. There exists a positive integer that is even.

iii. If x is even, then x is not divisible by 5.

iv. There exists an even integer divisible by 5.

A chi-square independence test is to be conducted to decide whether an association exists between two variables of a populations. One variable has 4 possible values and the other variable has 6. What are the degrees of freedom for the 𝜒2 -statistic?

answer:

(7 points) There is a popular story (among data miners) that there is a correlation between men buying diapers and buying beer while shopping. A student tests this theory by surveying 131 male shoppers as they left a grocery store. The results are summarized in the contingency table. Test for a dependent relationship between buying beer and buying diapers. Conduct this test at the 0.10 significance level.

Bought Diapers Did Not Buy Diapers Totals

Beer 7 46 53

No Beer 11 67 78

Totals 18 113 131

(a) Find the expected frequencies.

Bought Diapers Did Not Buy Diapers Beer

No Beer

(b) Find the test statistic.

(c) Find the critical value.