Question

this is probability and statistic topic in a contemporary math course.

please, need a full, accurate, and clear handwriting if you prefer to handwrite.

thanks in advance.  

2. The student council wants to know what activities would be supported by the student body. Decisions about activities are often made on the basis of survey information. Let's say that a survey was sent out to the student body via emails listed in the school's data base. The survey asked three questions: • Do you support the student council's decision to hold a fund raiser for the food bank? • Do you support the student council's decision to have a school social? Do you support the student council's decision to offer peer mentoring? Below is a Venn diagram showing the results of the students who responded to the survey. Use the diagram to answer the following questions. For each question show all calculations required and state your conclusion in a complete sentence or short paragraph. STUDENTS WHO RESPONDED TO SURVEY Fund Raiser Social 128 96 Mentoring 3,286 How many students responded to the survey? b. For the students who did support a decision or decisions, which single decision was supported by the most students? if the student council could make one single region of the Venn diagram larger (that is, make it have more members), which one would it be and why? d. How many students, who responded to the survey, didn't support any of the student councils decisions? Why do you think they did not support any of the decisions?

Answer 1

Let us denote the events of fund raising by A,School Social by B,Peer Monitoring by C, The event of voting is denoted by U.

Let n(X) denote the no. of students voted for the event X.

X$\cap$Y means the students who voted for both X & Y, for X,Y=A,B,C.

X$\cup$Y means the students who voted for either the event X or the event Y or both of the event.

n(A)=128+43+78+35=284

n(B)=35+78+96+52=261

n(C)=43+78+96+215=432

n(A$\cap$B)=35+78=113

n(A$\cap$C)=43+78=121

n(B$\cap$C)=96+78=174

n(A$\cap$B$\cap$C)=78.

n(U)=3286

a. n(U)=3286=The no. of students who responded in the survey.

b. Since n(C) is largest, So we see that the decision of peer monitoring is supported by most of the students.

c. The Student council could make the region of peer monitoring larger as it is supported by the maximum number of students who supported at least one decision.

d. By the Inclusion-Exclusion Principle,

n(A$\cup$B$\cup$C)=n(A)+n(B)+n(C)-n(A$\cap$B)-n(B$\cap$C)-n(A$\cap$C)+n(A$\cap$B$\cap$C)

=284+261+432-113-121-174+78

=1237

The Students who didn't support any of the student council's decision is

n(A$\cup$B$\cup$C)^c = n(U)- n(A$\cup$B$\cup$C) = 3286-1237=2049

They did not support any of the decision because they may be supporting any other cause like raising Sports Fund or Buy books for library, etc.