The data represent the age of world leaders on their day of inauguration. Find the​ five-number summary, and construct a boxplot for the data. Comment on the shape of the distribution.

49 57 62 55

43 68 58 64

43 57 56 65

48 48 50

The​ five-number summary is _____​,_______​,_______​,_______​,______.

​(Use ascending​ order.)

Choose the correct description of the shape of the distribution.

A. The distribution is roughly symmetric.The distribution is roughly symmetric.

B. The distribution is skewed to the left.The distribution is skewed to the left.

C. The distribution is skewed to the right.The distribution is skewed to the right.

D. The shape of the distribution cannot be determined from the boxplot.

An elevator has a placard stating that the maximum capacity is 2565 lb/15 passengers.​ So, 15 adult male passengers can have a mean weight of up to 2565 divided by 15 equals 171 pounds. If the elevator is loaded with 15 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 171 lb.​ (Assume that weights of males are normally distributed with a mean of 179 lb and a standard deviation of 27 lb ​.) Does this elevator appear to be​ safe?

Consider a continuous random vector (Y, X) with joint probability density function f(x, y) = 1 for 0 < x < 1, x < y < x + 1.

A. What is the marginal density of X and Y ? Use this to compute Var(X) and Var(Y).

B. Compute the expectation E[XY]

C. Use the previous results to compute the correlation Corr(Y, X).

D. Compute the third moment of Y , i.e., E[Y³].

1. Use ggplot R coding to create a scatterplot in order to investigate if there is a linear relationship between MidtermScore and FinalExamScore. Let MidtermScore be designated x and FinalExamScore be designated y.

2. Use R coding to find the correlation coefficient. Also, using two or three sentences, comment on the strength and the direction of the scatter plot. Would it be appropriate to apply a linear regression model to this data set?

3. Use R coding to find the coefficients of the Linear Regression Equation.  Write the Linear Regression (Least Squares Regression) equation.

4. Write an interpretation of the slope for the equation from problem 3.

5. Use R coding to produce the output table that gives the p values for each coefficient, the t values for each coefficient, the Standard error for each coefficient , and the Multiple-R Squared value.

6. Interpret the p values for each estimate in the table that you produced from problem 4.  Also interpret the Multiple-R Squared value in the table.

7. Use your equation from problem 3 to predict a Final Exam Score given a Midterm Score of 87.

8. Find the residual for the observational score of 80 and determine if that score is above or below average.  

. Construct a 98% confidence interval for the average monthly apartment rent using a random sample of 32 apartments with a mean of $375 and a standard deviation of 32.5.

A bottled water distributor wants to estimate the amount of water contained in 1​-gallon bottles purchased from a nationally known water bottling company. The water bottling​ company's specifications state that the standard deviation of the amount of water is equal to 0.02 gallon. A random sample of 50 bottles is​ selected, and the sample mean amount of water per 1​-gallon bottle is 0.995 gallon. Complete parts​ (a) through​ (d).

a. Construct a 95​% confidence interval estimate for the population mean amount of water included in a​ 1-gallon bottle

? ≤ μ ≤ ?

​(Round to five decimal places as​ needed.)

b. On the basis of these​ results, do you think that the distributor has a right to complain to the water bottling​ company? Why?

(Yes or No), because a​ 1-gallon bottle containing exactly​ 1-gallon of water lies (outside or within) the 95% confidence interval.

c. Must you assume that the population amount of water per bottle is normally distributed​ here? Explain. (Choose the answer below)

A. ​Yes, because the Central Limit Theorem almost always ensures that overbarX is normally distributed when n is large. In this​case, the value of n is small.

B. ​No, because the Central Limit Theorem almost always ensures that overbarX is normally distributed when n is large. In this​case, the value of n is large

C. ​No, because the Central Limit Theorem almost always ensures that overbarX is normally distributed when n is small. In this​case, the value of n is small.

D. ​Yes, since nothing is known about the distribution of the​population, it must be assumed that the population is normally distributed.

d. Construct a 90​% confidence interval estimate. How does this change your answer to part​ (b)?

? ≤ μ ≤ ?

​(Round to five decimal places as​ needed.)

How does this change your answer to part​ (b)?

A​ 1-gallon bottle containing exactly​ 1-gallon of water lies (outside or within) the 90​% confidence interval. The distributor (still has or now has or now does not have or still does not have) a right to complain to the bottling company.

1) Researchers wish to test the efficacy of a program intended to reduce the length of labor in childbirth. The accepted mean labor time in the birth of a first child is 15.3 hours. The mean length of the labors of 13 firsttime mothers in a pilot program was 8.8 hours with standard deviation 3.1 hours. Assuming a normal distribution of times of labor, test at the 10% level of significance test whether the mean labor time for all women following this program is less than 15.3 hours.

2) Six coins of the same type are discovered at an archaeological site. If their weights on average are significantly different from 5.25 grams then it can be assumed that their provenance is not the site itself. The coins are weighed and have mean 4.73 g with sample standard deviation 0.18 g. Perform the relevant test at the 0.1% (1/10th of 1%) level of significance, assuming a normal distribution of weights of all such coins.

3) An economist wishes to determine whether people are driving less than in the past. In one region of the country the number of miles driven per household per year in the past was 18.59 thousand miles. A sample of 15 households produced a sample mean of 16.23 thousand miles for the last year, with sample standard deviation 4.06 thousand miles. Assuming a normal distribution of household driving distances per year, perform the relevant test at the 5% level of significance.

4) The average number of days to complete recovery from a particular type of knee operation is 123.7 days. From his experience a physician suspects that use of a topical pain medication might be lengthening the recovery time. He randomly selects the records of seven knee surgery patients who used the topical medication. The times to total recovery were:

128,135,131,142,136,151,133.

a) Assuming a normal distribution of recovery times, perform the recovery test of hypotheses at the 10% level of significance.

b) Would the decision be same at the 5% level of significance? Answer either by constructing a new rejection region (critical value approach) or by estimating the p-value of the test in part(a) and comparing it.  

Random variable X is a continuous uniform (0,4) random variable and Y=X^(1/2). (Note: Y is always the positive root.)

What is the P[X>=E[X]] ?

What is the E[Y] ?

what is the P[Y>=E[Y]]?

what is the PFD of f_Y(y)?

The SUNY board of directors has 8 members.       

i. If they need to select a Chairman, a Secretary and a Treasurer, how many different slate of officers can be created?

ii.. If they need to select a 3 member subcommittee, how many different subcommittees can be formed?

b. A man pays $400 for a 1 year life insurance with coverage of $200,000. Given his age, he has a .995 chance of living for at least another year. what is the expected value of the policy?

Should he buy it or not? Why or Why not?

c. The TV show Scandal has a 20 share, meaning that 20% of the households are tuned to this show. A special focus group of 120 households are chosen.

i. What is the expected number of households watching this show?

ii. What is the mean number of households watching this show?

iii. What is the standard deviation?

  A furniture manufacturer produces two types of display cabinets Type A and Type B Each month x of type A and y of type B are produced. Profit on type A is 300SR and profit on type B is 150SR. The following constraints control monthly production :

(i)              Not more than 50 display cabinets of type A and 40 display cabinets of type B can be made

(ii)            To show a profit at least 60 display cabinets in all must be made.

(iii)           The maximum number of display cabinets that can be produced is 80.

How many display cabinets of type A must be produced per month to maximize the profit:

Write out the null and alternative hypotheses for the following hypothetical proposal. Carry out a one-sample Z test to determine significance at the α=0.05 level.

PROPOSAL

In the field of cancer epidemiology, many researchers are interested in developing risk measurement assays that are intended to influence the screening process for prevention of late stage and metastatic cancers. Parity, or the number of children that a woman has over her lifetime, is associated with the overall risk of breast cancer in women. Suppose that a researcher gathers a SRS of 135 women, and calculates an average of 3.2 births for the variable parity. Assume that women who give birth less than 4 times are at a 1.3 times increased risk for developing breast cancer. Does this researcher have sufficient evidence to claim that women in his population are at an increased risk for the disease? Use a population standard deviation of 0.37 births based on former research.

A regional planner employed by a public university is studying the demographics of nine counties in the eastern region of an Atlantic seaboard state. She has gathered the following data:

County

A.)Use regression analysis to determine the relationship between median income and median age. (Round your answers to 2 decimal places.)

Income=__________+_________Median Age

B.)Interpret the value of the slope in a simple regression equation. (Round your answers to 2 decimal places.)

For each year (increase/decrease) in age, the income increases ____________ on average.

"Durable press" cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. The breaking strength of untreated fabric is normally distributed with mean 52 pounds and standard deviation 1.8 pounds. The same type of fabric after treatment has normally distributed breaking strength with mean 24.1 pounds and standard deviation 1.8 pounds. A clothing manufacturer tests 3 specimens of each fabric. All 6 strength measurements are independent. (Round your answers to four decimal places.) (a) What is the probability that the mean breaking strength of the 3 untreated specimens exceeds 50 pounds? (b) What is the probability that the mean breaking strength of the 3 untreated specimens is at least 25 pounds greater than the mean strength of the 3 treated specimens?

An insurance company sells a policy to airline passengers. If a flier misses the purchased flight due to medical reasons, the policy gives $300 to the flier. Otherwise, there is no return. Records show that about 5% passengers miss flight due to illness. You buy the policy for your next flight. Select the correct statement.

(A) The expected value of the amount of money you receive is $15. (B) The variance of the amount of money you receive is $4275. (C) The amount of money you could receive is either $0 or $300. (D) All of the above are correct.