Q1. Let {Xt |t ∈ [0, 1]} be a stochastic process such that EX2 t < ∞ for all t ∈ [0, 1] which is strictly stationary. Show that it is stationary

. Q2. Let {Xt |t ∈ I} be strictly stationary. Prove or disprove that process is with stationary increments.

Q3. Let {Xt |t ∈ I} be with stationary increments. Prove or disprove that the process is stationary.

Q4. Prove or disprove that the stochastic process {Xn|n ≥ 0}, where {Xn} is i.i.d., is with independent and stationary increments.

Q5. Prove or disprove that SSRW is stationary.

11.3.73 Q20 Kevin installed a certain brand of automatic garage door opener that utilizes a transmitter control with four independent​ switches, each one set on or off. The receiver​ (wired to the​ door) must be set with the same pattern as the transmitter. If five neighbors with the same type of opener set their switches​ independently, what is the probability of at least one pair of neighbors using the same​ settings? The probability of at least one pair of neighbors using the same settings is approximately _____

A simple random sample of 60 items resulted in a sample mean of 90. The population standard deviation is σ = 17.

a. Compute the 95% confidence interval for the population mean. Round your answers to one decimal place.

b. Assume that the same sample mean was obtained from a sample of 120 items. Provide a 95% confidence interval for the population mean. Round your answers to two decimal places

According to a recent report, a sample of 360 one-year-old baby boys in the United States had a mean weight of 25.5 pounds. Assume the population standard deviation is =σ5.1 pounds.Construct a 99.5% confidence interval for the mean weight of all one-year-old baby boys in the United States. Round the answer to at least one decimal place.

Economists often track employment trends by measuring the proportion of people who are “underemployed,” meaning they are either unemployed or would like to work full time but are only working part-time. In the summer of 2019, 18.5% of Americans were “underemployed.” The mayor of Detroit wants to show the voters that the situation is not as bad in his city as it is in the rest of the country. His staff takes a simple random sample of 400 Detroit residents and finds that 60 of them are underemployed.  

(a) Does the data give convincing evidence that the proportion of underemployed in Detroit is lower than elsewhere in the country? Perform the appropriate statistical test.

(c) Suppose the true underemployment rate in Detroit is actually only 14%. If the mayor were to perform the exact same test again, what is the probability that the mayor’s test results in a Type II Error at the 5% level? Also Draw a graph to show it

Trading volume on the New York Stock Exchange is heaviest during the first half hour (early morning) and last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 17 days in January and February are shown here (Barron's, January 23, 2006; February 13, 2006; and February 27, 2006)

a. Compute the mean and standard deviation to use as estimates of the population mean and standard deviation

b. What is the probability that, on a randomly selected day, the early morning trading volume will be less than 195 million shares?

c. What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares?

d. How many shares would have to be traded for the early morning trading volume on a particular day to be among the busiest 5% of days?

**For B, C, D you show the bell curves in addition to the written work**

1. Find the SSE for the given data and linear models, and indicate which model gives the better fit.

(2,4) (6,8) (8,12) (10,0)

1. Y = - 0.1 x + 7                        SSE = ________
2. Y = - 0.2 x + 6                        SSE = ________
3. The better fit is          Y = __________________

2. The following table shows the average price of a two-bedroom apartment in downtown New York City from 1994 to 2004 (x=0 represents 1994)

find:

x = ______              y = ______             xysum of = ______          

x2 = ______            y2 = _______       

Regression line: ___________________________

Correlation Coefficient (2 decimal places): ____________

Using the regression line, what would be the price for 2007? ________

The extract of a plant native to Taiwan has been tested as a possible treatment for Leukemia. One of the chemical compounds produced from the plant was analyzed for a particular collagen. The collagen amount was found to be normally distributed with a mean of 72 and standard deviation of 5.1 grams per mililiter.
(a) What is the probability that the amount of collagen is greater than 63 grams per mililiter?
(b) What is the probability that the amount of collagen is less than 86 grams per mililiter?
（c)What percentage of compounds formed from the extract of this plant fall within 1 standard deviations of the mean?

On an installment of Who Wants to Be a Millionaire, a contestant had reached the $150,000 level. Confronted with a difficult question, he eliminated two of the wrong answers but still had no clue as to the correct answer. Guessing correctly would have increased his winnings by $150,000 (doubled) and kept him in the game while guessing it incorrectly would have reduced his winnings to only $50,000 (resulting in a $100,000 decrease in winnings).

a) If he elects to quit and keep the $150,00 is he considered risk-averse? Why?

b) What would you do and why?

The accompanying data are​ drive-through service times​ (seconds) recorded at a fast-food restaurant during dinner times. Assuming that dinner service times at the​ restaurant's competitor have standard deviation σ = 61.4​sec, use a 0.01 significance level to test the claim that service times at the restaurant have the same variation as service times at its​ competitor's restaurant. Use the accompanying data to identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, and​ P-value. Then state a conclusion about the null hypothesis. Need help with part B and beyond

Data: 144, 75, 122, 175, -42, 15, 16, 52, -5, -51, -107, -107

A) Identify null and alternative hypotheses (I already have answered, but gave just in case needed for rest).

H0: σ = 611.4 minutes

Ha σ *Doesn't equal* 61.4 minutes

B) Compute the test statistic (2 decimal places for rounding).

X^2 = _____

C) Find the P-Value of the test-statistic (3 decimal places for rounding).

The P-value for the test statistic is _______

D) State the conclusion about the null hypothesis

There _______ ("is" or "is not") sufficient evidence to conclude that there is a difference between the waiting times in the two​ restaurants, because H0 is _______ ("rejected" or "not rejected") by the hypothesis test.

Thank you in advance!

A small regional carrier accepted 16 reservations for a particular flight with 12 seats. 6 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 50% chance, independently of each other.

Find the probability that overbooking occurs.    
Find the probability that the flight has empty seats.  

1. Describe how to use a simple (bivariate) regression model to carry out a difference in the means test, to estimate a descriptive statistic, and to estimate an unbiased (or less biased) causal effect.

Heat treating is often used to carburize metal parts, such as gears. The thickness of the carburized layer is considered a crucial feature of the gear and contributes to the overall reliability of the part. Because of the critical nature of this feature, two different lab tests are performed on each furnace load. One test is run on a sample pin that accompanies each load. The other test is a destructive test, where an actual part is cross-sectioned. This test involves running a carbon analysis on the surface of both the gear pitch (top of the gear tooth) and the gear root (between the gear teeth). Table 12-6 shows the results of the pitch carbon analysis test for 32 parts.

1. Fit a regression model using all five regressors. Write out the equation. Summarize this analysis by providing the equation, ?², and list the standard errors.    What does the analysis of variance indicate?
2. Check for multicollinearity. Explain your findings.
3. Describe what you notice about the p values for the regressors. Construct a t-test on each regression coefficient. What can you conclude about the variables in this model? Use an alpha = 0.05.
4. Prepare a normal probability plot of the residuals and check the adequacy of the model.