3. A machine is used to fill containers with a liquid product. Fill volume can be assumed to be
normally distributed. A random sample of ten containers is selected, and the net contents
(oz) are as follows: 12.03, 12.01, 12.04, 12.02, 12.05, 11.98, 11.96, 12.02, 12.05, 11.99.
a. Suppose that the manufacturer wants to be sure       that the mean net contents
exceeds 12 oz. What conclusions can be drawn from the data. Use a = 0.01.
b. Construct a 95% two-sided confidence interval on the mean fill volume.
c. Does the assumption of normality seem appropriate for the fill volume data?
d. Repeat (a,b,c) above for a = 0.05. Compare with results for a = 0.01.

A particular county in a certain state of a far away country has 17 cities and wants to build roads between them to connect them all, that is, there should be a paved path between any two cities. What is the minimum number of roads, each connecting two cities, to guarantee that all cities are connected? Be careful, wrong answers have negative weights.

Denition:
An orthogonal array OA(k, n) on n symbols is an n² x k array such that, in any two columns, each ordered pair of symbols occurs exactly once.
Prove that there exists an OA(k, n) if and only if there exist (k - 2) mutually orthogonal Latin squares of order n.

(combinatorics and design)

Carlo and Anita make mailboxes and toys in their craft shop near Lincoln. Each mailbox requires 1 hour of work from Carlo and 2 hours from Anita. Each toy requires 1 hour of work from Carlo and 3 hours from Anita. Carlo cannot work more than 55 hours per week and Anita cannot work more than 12 hours per week. If each mailbox sells for $ 15 and each toy sells for $ 24, then how many of each should they make to maximize their​ revenue? What is their maximum​ revenue?

Carlo and Anita should make __ mailboxes and __ toys. Their maximum revenue is $__.

Use the Fourier-Motzkin Elimination method to solve problem 4.1-5 from the textbook (10th edition).

Maximize Z=x₁ + 2x₂,

subject to

x₁ + 3x₂ <=8

x₁ + x₂ <=4

and

x₁ >=0, x₂>=0.

A study was conducted concerning the effects of varying lengths of unemployment on ability to concentrate. Data represent participants' proofreading scores (number of errors found). Finding fewer errors on the proofreading task is a behavioral indicator of stress. Length of unemployment was regarded by the researchers as an indicator of exposure to chronic stress. Fifteen participants were included in the study and followed across three periods: employment (control), short-term unemployment and long-term unemployment. (20%) Data were collected and are shown below: Length of Unemployment

Control: 8 10 9 8 10. Short: 0 1 2 7 5 Long: 6 7 7 6 9

Please analyze the data fully (including appropriate follow-up tests) and provide a brief written summary of the results.

Determine all maximal planar graphs G of order 3 or more such that the number of regions in a planar embedding of G equals its order.

Give an example of two non-isomorphic maximal planar graphs of the same order.

FarmFresh Foods manufactures a snack mix called TrailTime by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below.

The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the TrailTime blend. TrailTime is packaged in boxes that will hold between three and four cups. The blend should contain no more than 1,100 calories and no more than 30 grams of fat. Dried fruit must be at least 25% of the volume (in cups) of the mixture, and nuts must be no more than 20% of the weight (in ounces) of the mixture.

Use Excel's Solver to obtain the optimal solution for this problem. In other words: what is the most cost effective way to create the TrailTime blend? What is the cost?

True or False? For any two events A and B, P(A∩ B) ≥ 1 − P(A∪ B)

True or False? For any two independent events A and B, P(A| B) = P(A| Bc )

Compute B(6, .2, 2)

Find the matrix A representing the follow transformations T. In each case, check that Av = T(v)
Step by step please.

A. T(x,y,z) = (x-3y+4z, 6x-2z, 8x-y-4z)

B. T(x,y) = (x,y,y-x,x+y, 6x-9y)

Thank you!

Find the general solution to the differential equation 3x''+2x'+2x=0. Use  and  to denote arbitrary constants.

6a. Show that 2/n = 1/3n + 5/3n and use this identity to obtain the unit fraction decompositions of 2/25 , 2/65 , and 2/85 as given in the 2/n table in the Rhind Mathematical Papyrus.

6b. Show that 2/mn = 1/ (m ((m+n)/ 2 )) + 1/ (n ((m+n)/ 2 )) and use this identity to obtain the unit fraction decompositions of 2/7 , 2/35 , and 2/91 as given in the 2/n table in the Rhind Mathematical Papyrus.

6c. Verify that 2/ n = 1/n + 1/2n + 1/3n + 1/6n and use this identity to obtain the unit fraction decompositions of 2/101 as given in the 2/n table in the Rhind Mathematical Papyrus.

IN EXCEL: 5.2.1 At time t = 0, a yeast culture weighs 0.5 g. Two hours later, it weighs 2 g. The maximum weight of the culture is 8 g.

1. Create a spreadsheet to model the population using a logistic equation. Use a scroll bar to vary the value of k.

2. Use the scroll bar to find a value of k so that the condition y(2) = 2 is satisfied.

3. At what time is the weight increasing most rapidly? Support your answer numerically.

Find the eigenvalues

λ_n

and eigenfunctions

y_n(x)

for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.)

x²y'' + xy' + λy = 0,  y'(e⁻¹) = 0,  y(1) = 0