6. Let V be the vector space above. Consider the maps T : V → V And S : V → V

defined by T(a1,a2,a3,...) = (a2,a3,a4,...) and S(a1,a2,a3,...) = (0,a1,a2,...).

(a) [optional] Show that T and S are linear.
(b) Show that T is surjective but not injective.
(c) Show that S is injective but not surjective.

(d) Show that V = im(T) + ker(T) but im(T) ∩ ker(T) ̸= {0}.

(e) Show that im(S) ∩ ker(S) = {0} but V ̸= im(S) + ker(S).

Sophie's given utility function is U(V, C) = VC + V where V = # of visits per month and C = # of minutes per month (hundreds). V = $1 and C = $4 and M = $44.

a) Calculate the equation that represents the consumers demand for number of visits (V).

b) Say that V increases to $16. Calculate the new equilibrium. Compute Sophie's substitution and income effect and represent it within a diagram.

Prove that (((p v ~q) ⊕ p) v ~p) ⊕ (p v ~q) ⊕ (p ⊕ q) is equivalent to p ^ q. Please show your work and name all the logical equivalence laws for each step. ( v = or, ~ = not, ⊕ = XOR)

Thank you

1. According to the National Institute on Alcohol Abuse and Alcoholism and the National Institute of Health, 41% of college students nation-wide engage in “binge drinking” behavior, that is having 5 or more drinks in one occasion during the past two weeks. A college president wonders if the proportion of students enrolled at her college who binge drink is different than the national proportion.

In a commissioned study of her college only, 348 students are randomly selected from a list of all students enrolled at the college. Of these, 118 admit to having engaged in binge drinking.

1. Calculate the sample proportion of binge drinkers randomly selected at this college for this sample.

2. Check the two necessary conditions to determine if it is appropriate to use normal distribution calculations in this case for this proportion problem.

3. Assuming the conditions above are met, calculate a 90% confidence interval for the proportion of all students at this college that engage in binge drinking (show all work). Interpret your results. Use standard error = 0.025.

4. Given that the national proportion of college binge drinkers is 0.41, should this college president be concerned the proportion of binge drinkers at her college are higher than the national proportion? Explain. Hint: think about the confidence interval end points and determine if the national proportion of binge drinkers, 0.41, is in that interval.

1. New York wants to do a study to compare students at two different schools in the city to see if student’s feelings about college are influenced by their tuition. The city takes a sample of 100 students from Barnard college whose tuition is 60 thousand dollars a year and 200 from Lehman College whose tuition is 15 thousand dollars a year. They ask each student the same question “Do you feel that your college is worth the tuition you pay for it?”. The survey found that 80% of the students at Lehman college agreed with the statement, while only 60% of the students at Barnard college agreed with the statement. The Research group believes that the proportion of students who agrees with the question will be the same between the two group. Assume that the data is normally distributed, and the sample was randomly selected. Please test this hypothesis and use an alpha of .05 for this study. Please answer the questions below. (points 10)
   1. What I the null and alternative hypotheses of the study? (2)
   2. What are the requirements for using a normal distribution to solve this problem? List them below (3)
   3. What is the critical value for this distribution? (1)
   4. What is the pooled probability of success for the study? (1)
   5. What is the sample statistics for this distribution? (1)
   6. Do I reject or accept the null hypotheses in my example? Explain? (1)

1. New York wants to do a study to compare students at two different schools in the city to see if student’s feelings about college are influenced by their tuition. The city takes a sample of 100 students from Barnard college whose tuition is 60 thousand dollars a year and 200 from Lehman College whose tuition is 15 thousand dollars a year. The ask each student the same question “Do you feel that your college is worth the tuition you pay for it?”. The survey found that 80% of the students at Lehman college agreed with the statement, while only 70% of the students at Barnard college agreed with the statement. The City believes that the proportion of students at Barnard and Lehman who think college is worth the money they pay for should be the same. Please test this hypothesis and use an alpha of .05 for this study. Please answer the questions below. (points 10)
   1. What Is the null and alternative hypotheses of the study? (2)
   2. What are the requirements for using a normal distribution to solve this problem? List them below (2)
   3. What is the critical value for this distribution? (1)
   4. What is the pooled probability of success for the study? (1)
   5. What is the standard deviation? (1)
   6. What is the sample statistics for this distribution? (1)
   7. Do I reject or accept the null hypotheses in my example? Explain? (2)

Austin Peay State, a university near Nashville, Tennessee, is applying a data-mining approach to higher education. Before students register for classes, a robot looks at their profiles and transcripts and recommends courses in which they are likely to be successful or have higher chances of success. The software takes an approach similar to the ones Netflix, eHarmony, and Amazon use to make their recommendations. It compares a student’s transcripts with those of past students who had similar grades and SAT scores. When a student logs in, the program offers 10 “Course Suggestions for You.” This recommendation is based on the student’s major and other information related to that student. The goal is to steer students toward courses in which they will make better grades. According to Tristan Denley, a former programmer turned math professor turned provost, students who follow the recommendations do substantially better. In the fall of 2011, 45 percent of the classes that students were taking had been on their top 10 recommendations list. This data-mining concept is catching on. Three other Tennessee colleges now use Denley’s software. Institutions outside the state are developing their own versions of the idea.

1. What are the benefits and drawbacks of this approach to course recommendations?
2. Are there any data that should or should not be included in data mining for this purpose? Why or why not?

The following table compares the percent sequence homology of four different parts (two introns and two exons) of a gene that is found in five different eukaryotic species. Each part is numbered to indicate its distance from the promoter (for example, Intron I is the one closest to the promoter). The data reported for species A were obtained by comparing DNA from one member of species A to another member of species A.

% Sequence Homology

Compute the minimum changes for each of the characters and list these below.

Intron I            _____________

Exon I              _____________

Intron VI          _____________

Exon V             _____________

Design and simulate a regulated power supply using a bridge rectifier, capacitors, and zener diode ( no integrated circuit). The source voltage is 110 plus or minus 10 Vrms, 60 Hz frequency. The output voltage is as follows (plus or minus 5%) : 4.5 V and 12 V

It has to be done on a breadboard. The rating of the adapter will be 12W and 5% regulation. To protect the circuit include a surge protection and fuse in the circuit design. The total cost of the components of the design should be less than $10. A total number of the elements should be less than 12 for reducing environmental impact. Test it on Multisim and breadboard. Please LIST each item that you used on the breadboard test exactly. I NEED YOU TO PERFORM IT ON A BREADBOARD as well as ON MULTISIM. SHOW the multilsim picture. SHOW ALL CALCULATIIONS NEATLY! DO NOT WRITE IN PEN OR CURSIVE! THANK YOU!

let v and u are random variables fvu(v,u)= e^-v-u for u ,v>=0 Z= e^v+u find fz(z) ?