Jake's Battery Company has two service departments, Maintenance and Personnel. Maintenance Department costs of $300,000 are allocated on the basis of budgeted machine-hours. Personnel Department costs of $100,000 are allocated based on the number of employees. The costs of operating departments A and B are $160,000 and $240,000, respectively. Data on budgeted maintenance-hours and number of employees are as follows:

a. Using the direct method, what amount of Maintenance Department costs will be allocated to Departments A and B, respectively? (Round up)

b. Using the step-down method, what amount of Maintenance Department costs will be allocated to Departments A and B, respectively if the service department with the highest percentage of interdepartmental support service is allocated first? (Round up)

c. Using the reciprocal method, what amount of Maintenance Department costs will be allocated to Departments A and B, respectively? (Round up)

*Asker notes* I am mostly confused on how to attain the percentages, especially the percentages for the cost of the personnel department to the maintenance department based on number of employees and the percent off budgeted-maintenance hours to the personnel department, especially because you need the percentages of these items in order to solve the reciprocal method with a linear equation.

the number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with a mean of 1252 chips and standard deviation 129 chips.

(a) what is the probability that a randomly selected bag contains between 1000 and 1500 chocolate chips, inclusive?

(b) what is the probability that a randomly selected bag contains fewer than 1125 chocolate chips?

(c) what proportion of bags contains more than 1175 chocolate chips?

(d) what is the percentile rank of a bag that contains 1475 chocolate chips?

(a) the probability that a randomly selected bag contains between 1000 and 1500 chocolate chips, inclusive is _. (round to four decimal places as needed.)

(b) the probability that a randomly selected bag contains fewer than 1125 chocolate chips is _. (round to four decimal places as needed.)

(c) the proportion of bags that contains more than 1175 chocolate chips is _. (round to four decimal places as needed.)

(d) a bag that contains 1475 chocolate chips is in the _th percentile. (round to the nearest integer as needed.)

2 .Define the probability density functions (pdfs) of the independent random variables X and Y as:

f_X(x)={ (1/5).(1-(x/10)), 0≤x≤10 f_{Y(y)={ (1/5).(y/10) ,} 0≤y≤10

0 , otherwise for both f_X(x) and f_Y(y)

(a) Suppose that you are playing a game where you want to maximize your number of points , and you can have your score be X or Y . Which would you choose?

(b) Suppose I am confused in part (a), so I flip an unfair coin, with P (“heads”) = 3/4 , P (“tails”) = 1/4 , to select which probability density function to draw my score from: if the coin is “heads”, I choose X; if the coin is “tails”, I choose Y . What is the probability that I score more than 5 points.

(c) Suppose generate both X and Y . What is the probability that Y is bigger than X? That is, what  is P(Y >X)?

Suppose that it is known that the number of minutes that the daily 1 pm train from NJ Station arrives late to destination Linden is a random variable X with density function

f(x) = {c/15^2(15^2-x^2), -15<x<15;
0, elsewhere.

(a) Determine the value of c so that f(x) is a valid probability density function.

(b) Explain why c must be a particular value. That is, explain why probability density functions must have the specific property that you used to solve part (a).

(c) Use the value of c found in part (a). Find the probability that the train arrives between 0 (perfectly on time) and 10 minutes late.

(d) Plot the graph of the probability density function, shade in the area between x = 0 and x = 10, and explain what your result in (c) means with respect to your graph.

(e) Find the expected value of X, the variance of X, and the standard deviation of X. Note that, again, you should use the value found for c here.

Problem from statistic.

1.Problem

A research project analyzes cognitive abilities of left-handed children. In a small scale pre-study, 6 children are selected randomly from a kindergarten group of 20 children. Assume that 10% of the children in this kindergarten group are left-handers.

1. Find the probability that the sample of 6 children includes exactly one left-handed child. Note: Define appro- priate random variables and their distributions.
2. Find the probability that the sample includes more than two left-handed children.

2.Problem

An internet mail order business sends out emails to current clients to advertise a new product. It is known from previous advertising campaigns that the probability of an order is 0.18. The number of current clients is n = 100, 000. The product price is 70 euros.

1. Calculate the expected sales for the campaign. Note: define appropriate random variables assuming indepen- dence of orders.
2. Calculate the probability that sales are greater than 1,274,980 euros.

Please use Cumulative Areas of the Standard Normal Distribution

A recent survey reported that 57% to 18 to 29 year olds in a certain country own tablets. Using the binomial distribution , complete parts a through e below.

a. What is the probability that in the next six​ 18- to​ 29-year-olds surveyed, four will own a​ tablet?

b. What is the probability that in the next six? 18- to? 29-year-olds surveyed, all six will own a? tablet?

c. What is the probability that in the next six? 18- to? 29-year-olds surveyed, at least four will own a? tablet?

d. What are the mean and standard deviation of the number of? 18- to? 29-year-olds who will own a tablet in a survey of? six?

e. What assumptions do you need to make in? (a) through? (c)? Select all that apply.

-The outcome of any observation is independent of the outcome of any other observation.

-The probability of an observation being classified as the event of? interest,

pi??, is constant from observation to observation.

-The outcome of any observation is dependent of the outcome of any other observation.

-Each observation is classified into one of two mutually exclusive and collectively exhaustive categories.

The Wall Street Journal reported some interesting statistics on the job market. One statistic is that 40% of all workers say they would change jobs for "slightly higher pay." In addition, 88% of companies say that there is a shortage of qualified job candidates. Suppose 16 workers are randomly selected and asked if they would change jobs for "slightly higher pay."

Appendix A Statistical Tables

*(Round your answer to 3 decimal places when calculating using Table A.2.)
**(Round your answer to 4 decimal places.)
***(Round your answer to 1 decimal place.)

a. What is the probability that nine or more say yes? *
b. What is the probability that three, four, five, or six say yes? *
c. If 13 companies are contacted, what is the probability that exactly 10 say there is a shortage of qualified job candidates? **
d. If 13 companies are contacted, what is the probability that all of the companies say there is a shortage of qualified job candidates? **
e. If 13 companies are contacted, what is the expected number of companies that would say there is a shortage of qualified job candidates? ***

1. The area under the curve must add up to one for

   	a.	all density functions.
   	b.	just one density function.
   	c.	no density function.
   	d.	a special group of density functions.

3 points   

QUESTION 2

1. If the mean of a normal distribution is negative,

   	a.	the variance must also be negative.
   	b.	the standard deviation must also be negative.
   	c.	a mistake has been made in the computations, because the mean of a normal distribution can not be negative.
   	d.	Standard deviation can be any number but it must be positive.

3 points   

QUESTION 3

1. For a normal distribution, a negative value of Z indicates

   	a.	a mistake has been made in computations, because z is always positive.
   	b.	the area corresponding to the z is negative.
   	c.	the z is to the right of the mean.
   	d.	a value that is below the mean.

3 points   

QUESTION 4

1. The probability density function refers to:

   	a.	probability function for a discrete random variable.
   	b.	probability function for a continuous random variable.
   	c.	probability function for either a discrete or a continuous random variable.
   	d.	not enough information

4. Bayes Theorem - One way of thinking about Bayes theorem is that it converts a-priori probability to a-posteriori probability meaning that the probability of an event gets changed based upon actual observation or upon experimental data. Suppose you know that there are two plants that produce helicopter doors: Plant 1 produces 1000 helicopter doors per day and Plant 2 produces 4000 helicopter doors per day. The overall percentage of defective helicopter doors is 0.01%, and of all defective helicopter doors, it is observed that 50% come from Plant 1 and 50% come from Plant 2. [8 points]

(i) The a-posteriori probability of defective helicopter doors produced by Plant 1 is:

a. 0.0025

b. 0.025%

c. 0.25%

d. 0.015%

Clustering [4 Points]   

(ii)K-Means clustering algorithm

a. Needs K-means++ to know the optimal location of the centroids

b. Needs K-means++ to know the optimal number of clusters

c. Is a supervised algorithm

d. Provides the optimal clustering of points even if the initialization is bad

New legislation passed in 2017 by the U.S. Congress changed tax laws that affect how many people file their taxes in 2018 and beyond. These tax law changes will likely lead many people to seek tax advice from their accountants (The New York Times). Backen and Hayes LLC is an accounting firm in New York state. The accounting firms believe that it may have to hire additional accountants to assist with the increased demand in tax advice for the upcoming tax season. Backen and Hayes LLC has developed the following probability distribution for x= number of new clients seeking tax advice.

a. Is this a valid probability distribution? Explain.

b. What is the probability that Backens and Hayes LLC will obtain 40 or more new clients?

c. What is the probability that Backens and Hayes LLC will obtain fewer than 35 new clients?

d. Compute the expected value, variance, and standard deviation of x.