The number of letters arriving each day at a residential address is assumed to be Poisson distributed with mean 1.8. The numbers of letters arriving on different days are independent random variables. (i) Calculate the probability that exactly two letters arrive at the address in one day. (ii) Calculate the probability that no more than 5 letters arrive at this address in a 5 day period. (iii) On a particular day, there are no letters at this address. Find the probability that exactly 6 days go by before this happens again. (iv) Use a suitable approximation to calculate the probability that during a 30 day period, more than 65 letters are received at this address, with mean rate λ = 1.8 for each day

HOW DO YOU DETERMINE IF ITS TRUE OR FALSE WHEN DEALING WITH BINOMINAL VARIABLES

a) If sample data are such that the null hypothesis is rejected at the alpha=5% level of significance based upon a 2 tailed test, is H0 also rejected at the alpha=1% level of significance ? Explain. b) If a 2 tailed hypothesis test leads to rejection of the null hypothesis at a certain level of significance, would the corresponding 1 tailed test lead to rejection of the null hypothesis ? Explain.

Complete each problem on a separate worksheet in a single Excel file. Rename the separate worksheets with the respective problem number. You may have to copy and paste the datasets into your homework file first. Name the file with your last name, first initial, and HW #2. Label each part of the question. When calculating statistics, label your outputs. Use the Solver add-in for these problems.

1. For a telephone survey, a marketing research group needs to contact at least 600 wives, 480 husbands, 400 single adult males, and 440 single adult females. It costs $3 to make a daytime call, and (because of higher labor costs) $5 to make an evening call. Problem #1 lists the results that can be expected. For example, 30% of all daytime calls are answered by a wife, 15% of all evening calls are answered by a single male, and 40% of all daytime calls are not answered at all. Due to limited staff, at most 40% of all calls can be evening calls.

1. Determine how to minimize the cost of completing the survey.
2. Use SolverTable to investigate changes in the unit cost of either type of call. Specifically, investigate changes in the cost of a daytime call, with the cost of an evening call fixed, to see when (if ever) only daytime calls or only evening calls will be made. Then repeat the analysis by changing the cost of an evening call and keeping the cost of a daytime call fixed.

On the basis of a physical examination, a doctor determines the probability of no tumour   (event labelled C for ‘clear’), a benign tumour (B) or a malignant tumour (M) as 0.7, 0.2 and 0.1 respectively.

A further, in depth, test is conducted on the patient which can yield either a negative (N) result or positive (P). The test gives a negative result with probability 0.9 if no tumour is present (i.e. P(N|C) = 0.9). The test gives a negative result with probability 0.8 if there is a benign tumour and 0.2 if there is a malignant tumour.

(i) Given this information calculate the joint and marginal probabilities and display in the table below.

2. Obtain the posterior probability distribution for the patient when the test result is

            a) positive,   b) negative

4. Comment on how the test results change the doctor’s view of the presence of a tumour.

The article “Application of Surgical Navigation to To-
tal Hip Arthroplasty” (T. Ecker and S. Murphy, Journal

of Engineering in Medicine, 2007:699–712) reports
that in a sample of 113 people undergoing a certain
type of hip replacement surgery on one hip, 65 of them
had surgery on their right hip. Can you conclude that
frequency of this type of surgery differs between right
and left hips?

(a) State the null and alternative hypotheses.

(b) Use a two-sided 95% Agresti confidence interval for the population proportion to conduct this hypoth-

esis test. After constructing your confidence interval, state the conclusion in the context of the study

and the conclusion of the hypothesis test.

(c) Go ahead and formally conduct the hypothesis test using a = 0.05. Calculate the test statistic and

P-value appropriately. Does your conclusion to the hypothesis test agree with part (b)?

3. It is thought that the mean length of trout in lakes in a certain region is 20

Inches. A sample of 46 trout from one particular lake had a sample mean of 18.5 inches and a sample standard deviation of 4 inches. Conduct a hypothesis test at the 0.05 level to see if the average trout length in this lake is less than mu=20 inches.

The college bookstore tells prospective students that the average cost of its textbooks is $52 with a standard deviation of $4.50. A group of smart statistics students thinks that the average cost is higher. In order to test the bookstore’s claim against their alternative, the students will select a random sample of size 100. Assume that the mean from their random sample is $52.80. Test at 10% significance level.

--> Perform a hypothesis test and state your decision.

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 69 professional actors, it was found that 39 were extroverts.

(a) Let p represent the proportion of all actors who are extroverts. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 95% confidence interval for p. (Round your answers to two decimal places.)

lower limit:

upper limit:

1. Find the mean and the Variance of the following sample data:

Discuss the following in 175 words: The concept of mean & standard deviations of probability distributions play a significant role in managerial decision-making.

A random sample of 200 cars has 18 that are green in color. What is the confidence interval for the true proportion of green cars that lie within a 95% confidence interval?

Discuss the following in 175 words: How normal distributions play a significant role in managerial decision-making.

High rent district:the mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is 2544.Assume the standard deviation is 483.A real estate firm samples 83 apartments.Use cumulative distribution table if needed.

What is the probability that the sample mean rent is greater than 2614? Round atleast 4 places

What is the probability that sample mean rent is between 2413 and 2513? Round atleast 4 places

Find the 65th percentile of the sample mean rent? Round 4 places

Would it be unusual if the sample mean were greater than 2610? Round 4 places

Let's focus on the relationship between the average debt in dollars at graduation (AveDebt) and the in-state cost per year after need-based aid (InCostAid).

a) Does a linear relationship between InCost Aid and AveDebt seem reasonable? Explain.

b) Are there any unusual cases in this sample? If yes, state which ones they are and how they may be affecting the least-squares model fit.