From weather records, it is known that it is rainy in a particular city 35% of the time. When it is rainy, there is heavy traffic 80% of the time, and when it is not rainy, there is heavy traffic 25% of the time

. a. What is the probability that on a random day, it’s not raining and there is heavy traffic?

b. What is the probability that there is heavy traffic on a random day?

c. What is the probability that on a day with no rain there is no heavy traffic?

d. What is the probability that on a day with heavy traffic it is not raining?

e. What is the probability that if there is no heavy traffic, it is raining?

Prove the claim at the end of the section about the Euclidean Algorithm and Fibonacci numbers. Specifically, prove that if positive naturals a and b are each at most F(n), then the Euclidean Algorithm performs at most n − 2 divisions. (You may assume that n > 2.)

Define the exponentiation operator on naturals recursively so that x0 = 1 and xS(y) = xy · x. Prove by induction, using this definition, that for any naturals x, y, and z, xy+z = xy · xz and xy·z = (xy

A polygon is called convex if every line segment from one vertex to another lies entirely within the polygon. To triangulate a polygon, we take some of these line segments, which don’t cross one another, and use them to divide the polygon into triangles. Prove, by strong induction for all naturals n with n ≥ 3, that every convex polygon with n sides has a trian-gulation, and that every triangulation contains exactly n − 2 triangles. (Hint: When you divide an n-gon with a single line segment, you create an i-gon and a j-gon for some naturals i and j. What does your strong inductive hypothesis tell you about triangulations of these polygons?)

QUESTION 3 [20 MARKS]

a. State any five (5) factors that often restrict the sample size.

b. Explain with reasons the most suitable measurement scale that a researcher can use from these sampled statements: i. Female soccer teams in this year’s Olympics. ii. Number of countries in Africa. iii. Seasons in a year.

c. Distinguish Secondary literature from Primary literature use examples to justify your answer.

Rita is the owner of Rita’s Osteria. She wants to study the growth of her business using simulation. She is interested in simulating the number of customers and the amount ordered by customers each month. She feels that the number of customers is normally distributed, with a mean of 800 and a standard deviation of 45. The bill for each customer is $65 and uniformly distributed, with a maximum increase of 8% and a minimum decrease of 5%. The changes of the bills are incremental, i.e. each change is based off the average bill of the previous month. Formulate a simulation, computing the mean total revenue and the standard deviation in one year.

Because there are infinitely many primes, we can assign each one a number: p0 = 2, p1 = 3, p2 = 5, and so forth. A finite multiset of naturals is like an ordinary finite set, except that an element can be included more than once and we care how many times it occurs. Two multisets are defined to be equal if they contain the same number of each natural. So {2, 4, 4, 5}, for example, is equal to {4, 2, 5, 4} but not to {4, 2, 2, 5}. We define a function f so that given any finite multiset S of naturals, f(S) is the product of a prime for each element of S. For example, f({2, 4, 4, 5} is p2p4p4p5 = 5 × 11 × 11 × 13 = 7865.
(a) Prove that f is a bijection from the set of all finite multisets of naturals to the set of positive naturals.
(b) The union of two multisets is taken by including all the elements of each, retaining du-plicates. For example, if S = {1, 2, 2, 5} and T = {0, 1, 1, 4}, S∪T = {0, 1, 1, 1, 2, 2, 4, 5}. How is f(S ∪ T) related to f(S) and f(T)?
(c) S is defined to be a submultiset of T if there is some multiset U such that S ∪U = T. If S ⊂ T, what can we say about f(S) and f(T)?
(d) The intersection of two multisets consists of the elements that occur in both, with each element occurring the same number of times as it does in the one where it occurs fewer times. For example, if S = {0, 1, 1, 2} and T = {0, 0, 1, 3}, S ∩ T = {0, 1}. How is f(S ∩ T) related to f(S) and f(T

Assume that women have heights that are normally distributed with a mean of 64.1 inches and a standard deviation of 2.5 inches. Find the 85th percentile of women's heights.

180 students were asked to randomly pick one of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The number 7 was picked by 61 students.

a) For the sample, calculate the proportion of students who picked 7. (Round the answer to three decimal places.)

(b) Calculate the standard error for this sample proportion. (Round the answer to three decimal places.)

(c) Calculate a 90% confidence interval for the population proportion. (Round the answer to three decimal places.) to

(d) Calculate a 95% confidence interval for the population proportion. (Round the answer to three decimal places.) to

(e) Calculate a 98% confidence interval for the population proportion. (Round the answer to three decimal places.) to

(f) What do the results of parts (c)-(e) indicate about the effect of confidence level on the width of a confidence interval? (Select all that apply.)

As the confidence level is increased, the width of the interval decreases.

As the confidence level is decreased, the width of the interval decreases.

As the confidence level is decreased, the width of the interval increases.

As the confidence level is increased, the width of the interval increases.

(g) On the basis of these confidence intervals, do you think that students choose numbers "randomly"?

Yes

No

1. Five cards are dealt at random from a well-shuffled deck of 52 playing cards. Find the probability that: a. All are spades. b. Exactly two are hearts. c. Exactly three are clubs. d. All are red. e. At least one card is ace. 2. Tossing a coin 15 times find the probability of getting exactly 4 tails. 3. Find the probability of getting at least 4 tails for tossing a coin 15 times

In a recent year, the Better Business Bureau settled 73% of complaints they received. You have been hired by the Bureau to investigate complaints this year involving computer stores. You plan to select a random sample of complaints to estimate the proportion of complaints the Bureau is able to settle. Assume the population proportion of complaints settled for the computer stores is 0.73, as mentioned above. Suppose your sample size is 190. Please answer the following questions, and show your answers to 4 decimal places if possible.

a. What is the distribution of ˆpp^? ˆpp^ ~ N( , )  

b. What is the probability that the sample proportion will be within 4 percent of the population proportion?

Let X∼Binomial(n,p) and Y∼Bernoulli(p) be independent random variables. Find the distribution of X+Y using the convolution formula

The Center for the Display of Visual Arts wants to know whether its parking situation is getting worse. Specifically, administrators are concerned that a lack of parking is driving away potential visitors. The administrators plan to address the situation if at least 15% of visitors feel that finding parking near the facility is a problem (the center averages about 3,100 visitors per week). To get a sense of whether this is really the case, the administrators distribute a brief exit survey to 150 individuals who visit the center over the following week. The results indicate that 32 of 150 visitors complained about being unable to find adequate parking. Present a hypothesis and null hypothesis. Calculate a t score and use it to evaluate the hypotheses. What should the administrators conclude from these results?

8. (This problem is required to submit) Bank of America’s Consumer Spending Survey collected data on annual credit card changes in seven different categories of expenditures: transportation, groceries, dining out, household expenses, home furnishings, apparel, and entertainment. Using data from a sample of 42 credit card accounts, assume that each account was used to identify the annual credit card charges for groceries (population 1) and the annual credit card charges for dining out (population 2). Using the difference data, the sample mean difference was 850 $d (Note: d = difference = groceries – dining out charges) and the same standard deviation was 1123 $ds . d = Difference = groceries charges – dining out charges. 10

a. Use math symbol to formulate the null and alternative hypotheses to test for no difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out. (10%)

b. Use a 0.05 level of significance. What is the p-value? Use the p-value approach, can you conclude that the population means differ? (10%) (Note: without calculation process, explanation, and conclusion, no credit)

c. Which category, groceries or dining out, has a higher population mean annual credit card charge? What is the point estimate of the difference between the population means? What is the 95% confidence interval estimate of the difference between the population means? (10%) (Note: without calculation process and explanation, no credit)

A government sample survey plans to measure the LDL (bad) cholesterol level of an SRS of men aged 20 to 34.
Suppose that in fact the LDL cholesterol level of all men aged 20 to 34 follows the Normal distribution with mean

μ = 119 milligrams per deciliter (mg/dL) and standard deviation σ = 25 mg/dL. Use Table A for the following questions, where necessary.

(a) Choose an SRS of 100 men from this population. What is the sampling distribution of x? (Use the units of mg/dL.)
Answer = N(119,2.5)

(b) What is the probability that x takes a value between 116 and 122 mg/dL? This is the probability that x estimates μ within ±3 mg/dL. (Round your answer to three decimal places.)

I only need the answer to part (b) I asked this question before and the answer is NOT 0.096