A survey of the mean number of cents off that coupons give was conducted by randomly surveying one coupon per page from the coupon sections of a recent San Jose Mercury News. The following data were collected: 20¢; 75¢; 50¢; 65¢; 30¢; 55¢; 40¢; 40¢; 30¢; 55¢; $1.50; 40¢; 65¢; 40¢. Assume the underlying distribution is approximately normal. You wish to conduct a hypothesis test (α = 0.05 level) to determine if the mean cents off for coupons is less than 50¢.

1. State the null and alternate hypotheses clearly.
2. Conduct the hypothesis test based on the test statistic and critical value(s). Clearly indicate each.
3. What is the p-value? Use the p-value to conduct the same test
4. Report your conclusion in words, in the context of the problem.
5. What is the power of the for an alternative hypothesis value of 49¢?

If x is a binomial random variable, compute P(x) for each of the following cases:

(a)  P(x≤5),n=9,p=0.7P(x≤5),n=9,p=0.7

(b)  P(x>1),n=9,p=0.1P(x>1),n=9,p=0.1

(c)  P(x<3),n=5,p=0.6P(x<3),n=5,p=0.6

(d)  P(x≥1),n=6,p=0.9P(x≥1),n=6,p=0.9

Using the info below, answer the next following questions:

A survey of the mean number of cents off that coupons give was conducted by randomly surveying one coupon per page from the coupon sections of a recent San Jose Mercury News. The following data were collected: 20¢; 75¢; 50¢; 65¢; 30¢; 55¢; 40¢; 40¢; 30¢; 55¢; $1.50; 40¢; 65¢; 40¢. Assume the underlying distribution is approximately normal.

(a) Determine the sample mean in cents (Round to 3 decimal places)

(b) Determine the standard deviation from the sample . (Round to 3 decimal places)

(e) Construct a 95% confidence interval for the population mean worth of coupons.  Use a critical value of 2.16 from the t distribution.

What is the lower bound? ( Round to 3 decimal places )

(f)  Construct a 95% confidence interval for the population mean worth of coupons .

What is the upper bound? ( Round to 3 decimal places )

4. (a) In a fraud detection system a number of different algorithms are working indepen- dently to flag a fraudulent event. Each algorithm has probability 0.9 of correctly detecting such an event. The program director wants to be make sure the system can detect a fraud with high probability. You are tasked with finding out how many different algorithms need to be set up to detect a fraudulent event. Solve the following 3 problems and report to the director. [Total: 18 pts] (b) Suppose n is the number of algorithms set up. Derive an expression for the probability that a fraudulent event is detected. (6 pts) (c) Using R, draw a plot of the probability of a fraudulent event being detected versus n, varying n from 1 to 10. (6 pts) (d) Your colleague claims that if the company uses n = 4 algorithms, the probability of detecting the fraudulent event is 0.9999. The director is not convinced. Generate 1 million samples from Binomial distribution with n = 4, p = 0.90 and count the number of cases where Y = 0. Report the number to the director. (6 pts)

Thoroughly answer the following questions:

What is the difference between prevalence and incidence? Provide an example of each. Do not provide the definitions, explain in your own words.

The weights of 22 randomly selected mattresses were found to have a standard deviation of 3.17. Construct the 95% confidence interval for the population standard deviation of the weights of all mattresses in this factory. Round your answers to two decimal places.

1. A social psychologist wanted to determine whether attitudes of men toward abortion were different in rural and urban areas. He prepared a questionnaire and administered it to a group of city dwellers and a group of country dwellers. Each city man was matched with a country man on age, income, and education. High scores indicate positive attitudes towards abortion. The data:

1. What can the social psychologist say about differences in the attitudes of rural and urban men?
2. Justify the statistical method you used to analyze the data.

Nine experts rated two brands of Colombian coffee in a taste-testing experiment. A rating on a 7-point scale ( 1=1= 1 equals extremely unpleasing, 7=7= 7 equals extremely pleasing) is given for each of four characteristics: taste, aroma, richness, and acidity. The following data stored in Coffee contain the ratings accumulated over all four characteristics:

• a. At the 0.05 level of significance, is there evidence of a difference in the mean ratings between the two brands?

• b. What assumption is necessary about the population distribution in order to perform this test?

• c. Determine the p-value in (a) and interpret its meaning.

• d. Construct and interpret a 95% confidence interval estimate of the difference in the mean ratings between the two brands.

SHOW EXCEL FUNCTIONS USED TO ANSWER.

Isabella gathered data on the average percentage of tips received by waitstaff in 31 restaurants in New York City. She works through the testing procedure:

• H0:μ=14%; Ha:μ≠14%
  
• α=0.1 (significance level)
  
• The test statistic is t0=x¯−μ0sn√=1.489.
  
• The critical values are ±t0.05=±1.697.
  

Conclude whether to reject or not reject H0. Select two responses below.

Select all that apply:

• Reject H0.

• Fail to reject H0.

• The test statistic falls within the rejection region.

• The test statistic is not in the rejection region.

2. Set up both the vector of state probabilities and the matrix of transition probabilities given the following information: Store 1 currently has 40% of the market; store 2 currently has 60% of the market. In each period, store 1 customers have an 80% chance of returning; 20% of switching to store 2.

In each period, store 2 customers have a 90% chance of returning; 10% of switching to store 1. a.

Find the percentage of market for each store after 2 periods. b. Find the equilibrium conditions of 2 stores (limiting probabilities). What’s the meaning of these probabilities?

A study of the career paths of hotel general managers sent questionnaires to an SRS of 240 hotels belonging to major U.S. hotel chains. There were 133 responses. The average time these 133 general managers had spent with their current company was 12.37 years. (Take it as known that the standard deviation of time with the company for all general managers is 1.5 years.) (a) Find the margin of error for a 90% confidence interval to estimate the mean time a general manager had spent with their current company: years (b) Find the margin of error for a 99% confidence interval to estimate the mean time a general manager had spent with their current company: years (c) In general, increasing the confidence level the margin of error (width) of the confidence interval. (Enter: ''DECREASES'', ''DOES NOT CHANGE'' or ''INCREASES'', without the quotes.)

A) estimate the error in the values of the gaussian approximation of the binomial coefficients g(12,2s) as 2s changes from 0 to its maximum value. (N=12 2s between states)

B) How will the error in the value g(N,0) calculated using the gausian approximation in A if you use N=20?

Peak expiratory flow (PEF) is a measure of a patient’s ability to expel air from the lungs. Patients with asthma or other respiratory conditions often have restricted PEF. The mean PEF for children free of asthma is 306. An investigator wants to test whether children with chronic bronchitis have restricted PEF. A sample of 40 children with chronic bronchitis is studied, and their mean PEF is 279 with a standard deviation of 71. Is there statistical evidence of a lower mean PEF in children with chronic bronchitis? (α = 0.05, enter 1 for “yes”, and 0 for “no”).

A minority representation group accuses a major bank of racial discrimination in its recent hires for financial analysts. Exactly

16%

of all applications were from minority members, and exactly

14%

of the

2100

open positions were filled by members of the minority.

1. Find the mean of

   p

   , where

   p

   is the proportion of minority member applications in a random sample of

   2100

   that is drawn from all applications.
2. Find the standard deviation of

   p

   .
3. Compute an approximation for

   P≤p0.14

   , which is the probability that there will be

   14%

   or fewer minority member applications in a random sample of

   2100

   drawn from all applications. Round your answer to four decimal places.