Kindly assist me to solve the following

Question 3

To estimate the population mean weight of out-going packages, a shipping company randomly samples 70 packages and finds a mean weight of 52 pounds. The population standard deviation is known to be 10.1 pounds.

Construct the 88% confidence interval and state the lower bound ___________________ and upper bound ___________________ , each rounded to the nearest thousandths place.

Interpret your CI in context ___________________.

Question 4

A landscape engineer claims that the average area of a pond is at most 150 sqft. To test this claim 30 randomly selected ponds show a mean of 151.7 sqft. The population standard deviation is known to be 6.8 sqft. Use the pvalue method to test the claim at the 7% significance level.

a. State Ho and Ha ___________________

b. Determine Critical Value and identify the rejection region ___________________

c. State the test menu and determine Test Statistic ___________________

d. Determine the p-value ___________________

e. State the conclusion, explain why, and interpret in context ___________________

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3. (a) During lunch hour, arrivals of customers at a pizza hut restaurant follows a Poisson process with the rate of 120 customers per hour. The restaurant has one line, with three workers taking food orders at independent services stations. Each worker takes an exponen- tially distributed amount of time-on average 1 minute-to serve a customer. Let X, denote the number of customers in the restaurant (in line and being serviced) at time f. Then, the process (X, : 1 2 0) is a continuous-time Markov chain. (i) Show that the process is a birth-and-death process by giving the birth and death rates. [4 marks] (ii) Find the generator matrix of the above process. [5 marks ] (iii) For each integer & 2 0, derive the long-term probability that there are & customers in the restaurant. [8 marks ] (iv) Calculate the long-term probability that all three workers are busy. [6 marks ] (v) Find the average number of customers in the restaurant in the long term. [7 marks ] (b) A student support center has 3 tutors who help students with their home work. Students arrive at the center according to a Poisson process at rate A = 3 per hour. Each tutor's service time is exponentially distributed with average of 1/10 hours. Tutors' service times and student arrival times are independent. If all the tutors are busy when a student arrives at the center, the student will leave. Let X, denote the number of tutors who are busy at time t. Determine its generator matrix and stationary distribution. [10 marks ]Show that if the detailed balance equations for the B&D process are satisfied, then the limiting distribution equations (6.35) are satisfied. Show that this is true in general, that is if there is the solution to the detailed balance equations, then the stationary equations 0 = > Wk qkj - "j lj, j = 0, 1, 2, .. . are satisfied. (c) An example where we have a stationary solution, but the detailed balance equations have no solution, is given by the infinitesimal matrix 3 A = 2 Find the stationary distribution for this chain and show that the detailed balance equa- tions have no solution. Can you give verbal description of how this chain moves in the long run? How is this compared to the interpretation at the end of question (a)?QUESTION 2 what is the main difference between a paired-sample T-test and a one-sample T-Test? ) 1-sample t-test compares one sample mean to a null hypothesis value. A paired t-test calculates the difference between paired observations Oa 1-sample t-test compares one sample mean to another sample mean. A paired t-test calculates the difference between paired observations Q 2 1-sample t-test compares one sample mean to a null hypothesis value. A paired t-test calculates the difference between multiple null hypotheses () They are the same exact things. just referred to as different tests depending on what types of samples are drawn from the population31 Starting with the R function for the simulation of the immegration -death process, modify it to inclube " birth's in addition to immigrations and death Use your modified function to simulate a bug realisation from the birth-immigration- death process where the birth, death, and immigration rates are all one, starting from an initial condition of zero