It has been suggusted that the highest priority of retirees is travel. Thus, a study was conducted to investigate the differences in the length of stay of a trip for pre- and post-retirees. A sample of 691 travelers were asked how long they stayed on a typical trip. The observed results of the study are found below.

With this information, construct a table of estimated expected values.

Now, with that information, determine whether the length of stay is independent of retirement using α=0.01.

(a)  χ2=

(b) Find the degrees of freedom:

(c) Find the critical value:  

It has been suggusted that the highest priority of retirees is travel. Thus, a study was conducted to investigate the differences in the length of stay of a trip for pre- and post-retirees. A sample of 688 travelers were asked how long they stayed on a typical trip. The observed results of the study are found below.

Number of Nights / Pre-retirement / Post-retirement / Total

4−7 / 236 / 167 / 403

8−13 / 81 / 60 / 141

14−21 / 33 / 55 / 88

22 or more / 19 / 37 / 56 /

Total / 369 / 319 / 688

With this information, construct a table of estimated expected values.

Number of Nights / Pre-retirement / Post-retirement

4−7 / 216.1438 / 186.8561

8−13 / 75.6235 / 65.3764

14−21 / 47.1976 / 40.8023

22 or more / 30.0348 / 25.9651

Now, with that information, determine whether the length of stay is independent of retirement using α=0.05.

What is χ2=________

Two advertising media are being considered for the promotion of a product. Radio ads cost $480 each, while newspaper ads cost $550 each. The total budget is $20,000 per week. The total number of ads should be at least 30, with a max of 5 newspaper ads. Each newspaper ad reaches 8,000 people, while each radio ad reaches 5,000 people.

Let R = # of radio ads

Let P = # of newspaper ads

Max 5000 R + 8000P

s. t.

480R + 550P <= 20000 cost of ads

R + P >= 30 total # of ads

P <= 5 max number of newspaper ads

R,P >= 0

Round your answers to the highest whole numbers

The company wishes to reach as many people as possible while meeting all the constraints stated, what is maximum reach?

How many ads of each type should be placed?

Using Miller-Rabin primality test algorithm to write a Java program which can test if an integer is a prime. The input of the algorithm is a large positive integer. The output is “the number *** is a prime” or “the number *** is not a prime”. The error probability of the algorithm should be no more than 1 256 . Use this program to test some big integers. In Java, there is a class BigInteger. You can use methods of that class except the method isProbablePrime. write Java program.

Find the probability (approximate) in the following cases:
a) Of 1000 random digits, 7 does not appear more than 968 times
b) Rolling 12000 a die, the number of six is ​​between 1900 and 2150.
c) In 182 days, the number of units demanded of a certain product exceeds 6370 units (Note: the daily demand of the product has average 30 and standard deviation 6, and the independence of the demand of each day is assumed with respect to the rest)

According to a 2017 Wired magazine article, 60% of emails that are received are tracked using software that can tell the email sender when, where, and on what type of device the email was opened (Wired magazine website). Suppose we randomly select 10 received emails.

10. What is the expected number of these emails that are tracked?

11. What are the variance and standard deviation for the number of these emails that are tracked?

12. What is the probability that at least 8 emails are tracked?

The number of University graduates in a town is estimated to follow
a Binomial distribution with the probability of success p = 0.6. To
test the null hypothesis a random sample of 15 adults is selected. If
the number of graduates in the sample is between 6 and 12 inclusive,
we shall accept the null hypothesis to be p = 0.6, otherwise we shall
conclude that p 6= 0.6. Use the normal approximation to the binomial
distribution to

(ii) Find the power of the test for p = 0.5

Consider an experiment where fair die is rolled and two fair coins are flipped. Define random variable X as the number shown on the die, minus the number of heads shown by the coins. Assume that all dice and coins are independent.

(a) Determine f(x), the probability mass function of X

(b) Determine F(x), the cumulative distribution function of X (write it as a function and draw its plot)

(c) Compute E[X] and V[X]

Q7 A fair coin is tossed three times independently: let X denote the number of heads on the first toss (i.e., X = 1 if the first toss is a head; otherwise X = 0) and Y denote the total number of heads.

Hint: first figure out the possible values of X and Y , then complete the table cell by cell.

Marginalize the joint probability mass function of X and Y in the previous qusetion to get marginal PMF’s.

The Wall Street Journal Corporate Perceptions Study 2011 surveyed readers and asked how each rated the quality of management and the reputation of the company for over 250 worldwide corporations. Both the quality of management and the reputation of the company were rated on an excellent, good, and fair categorical scale. Assume the sample data for 200 respondents below applies to this study.

1)Use a 0.05 level of significance and test for independence of the quality of management and the reputation of the company.

-State the null and alternative hypotheses.

- Find the value of the test statistic

-Find the p-value

-State your conclusion.;

a)Reject H₀. We conclude that the rating for the quality of management is independent of the rating for the reputation of the company.

b)Reject H₀. We conclude that the rating for the quality of management is not independent of the rating for the reputation of the company.     

c)Do not reject H₀. We cannot conclude that the ratings for the quality of management and the reputation of the company are not independent.

d)Do not reject H₀. We cannot conclude that the rating for the quality of management is independent of the rating of the reputation of the company.

2)If there is a dependence or association between the two ratings, discuss and use probabilities to justify your answer.

For companies with an excellent reputation, the largest column probability corresponds to  ---Select--- excellent good fair management quality. For companies with a good reputation, the largest column probability corresponds to  ---Select--- excellent good fair management quality. For companies with a fair reputation, the largest column probability corresponds to  ---Select--- excellent good fair management quality. Since these highest probabilities correspond to  ---Select--- the same different ratings of quality of management and reputation, the two ratings are