Analyzing Survey Data In a survey of 100 investors in the...

Analyzing Survey Data In a survey of 100 investors in the stock market, 50 owned shares in IBM 40 owned shares in AT\&T 45 owned shares in GE 20 owned shares in both IBM and GE 15 owned shares in both AT\&T and GE 20 owned shares in both IBM and AT\&T 5 owned shares in all three (a) How many of the investors surveyed did not have shares in any of the three companies? (b) How many owned just IBM shares? (c) How many owned just GE shares? (d) How many owned neither IBM nor GE? (e) How many owned either IBM or AT\&T but no GE?

Analyzing Survey Data In a survey of 100 investors in the stock market,
50 owned shares in IBM
40 owned shares in AT\&T
45 owned shares in GE
20 owned shares in both IBM and GE
15 owned shares in both AT\&T and GE
20 owned shares in both IBM and AT\&T
5 owned shares in all three
(a) How many of the investors surveyed did not have shares in any of the three companies?
(b) How many owned just IBM shares?
(c) How many owned just GE shares?
(d) How many owned neither IBM nor GE?
(e) How many owned either IBM or AT\&T but no GE?

Key Concepts

Complement of a Set

The complement of a set encompasses all the elements in the universal set that are not in the specified subset. In survey problems, calculating the complement helps determine how many subjects do not satisfy a given condition, which is essential for answering questions about exclusion in overlapping groups.

Set Theory

Set theory is the branch of mathematics that deals with collections of objects, or sets, and the operations that can be performed on them, such as unions, intersections, and complements. It provides the fundamental framework for solving problems that involve grouping elements based on shared or distinct characteristics.

Inclusion-Exclusion Principle

This is a counting method used to find the size of the union of multiple sets by adding the sizes of the individual sets, subtracting the sizes of all their pairwise intersections, and then adding back the sizes of the triple intersections (and so on). This principle helps to accurately account for elements that appear in more than one set by correcting overcounts.

Venn Diagrams

Venn diagrams are graphical representations of sets and their relationships. They illustrate the overlaps between sets, making it easier to identify common elements (intersections) as well as unique elements, which is particularly useful when analyzing survey data and other problems involving unions and intersections of groups.

Transcript

00:01 So we are given a lot of information for this problem that we're expected to use.

00:05 And to help simplify some things, we're just going to start assigning variables to certain parts of the problem.

00:12 So we're going to assign a to ibm, b to a t, and t, and c to g.

00:21 So we know that a is equal to 50.

00:25 We know that b is equal to 40 and c is equal to 45.

00:33 We're also aware that a intersect c is equal to 20.

00:46 We're given that b intersect c is equal to 15.

00:53 And we're also given that a intersect b is equal to 20 as well.

01:01 Except to b is equal to 20.

01:03 And that a intersect b intersect c is equal to 5.

01:12 And we're expected to come up with a lot of information for this.

01:15 So for the first part, we're trying to find out how many of the 100 investors are not represented in this data given.

01:23 And to do that, we are trying to find out how many are represented.

01:29 That's an easier way to think of it.

01:31 So we're trying to find out a, union, b, union c.

01:35 And then we can take 100 minus this number, and now we'll represent everyone who's not there.

01:41 Because we know that a, union, b, union c is equal to the third.

01:46 Three elements added together minus the separate intersections.

01:53 So a plus b plus c, minus a intersect b, minus a intersect c, minus a intersect c, and then plus the intersection.

02:04 So plus a intersect b intersect c.

02:09 And the nice part of writing things out in more in terms of unions intersections as seen above, is that when we write an equation like this, we can know what we have to find if we're going to use this equation.

02:22 And in this case, we actually already have all the numbers for this equation.

02:27 So we can just start plugging in.

02:29 So this tells us that a, union b, union c, is equal to 50 plus 40, plus 45, minus 20, minus 20, minus 15, plus 5.

02:47 And this is going to be equal to 85.

02:54 So 85 investors are represented or 85 investors did invest in ibm, at &t, ng and that tells us that since there's 100 investors total, about 15 of them, 100 minus 85, 15 of them do not have any representation in those three companies, any stocks.

03:17 The next thing we're asked to find is how many own just shares in ibm.

03:25 So to calculate the people who only have shares in ibm, we can take the, we'll call that b, or a, because that's our a variable.

03:35 We can take a, let me just write that in the black.

03:43 So you can take a minus the intersections.

03:49 So a intersect b plus a intersect c minus a intersect b intersect c.

04:03 And so this will be equal to a, which is equal to 50, 50 minus 20 plus 20 minus 5 or 35.

04:17 So 50 minus 35 is equal to 15.

04:20 So there are 15 people who have just ibm shares.

04:30 And then we knew the same thing for the next couple parts.

04:34 Of how many people own just ge shares.

04:38 And that would be equal to e minus the b intersects.

04:42 So a intersect b plus b intersect c minus a intersect b intersect c...