Chapter 13.1, Problem 29AYU

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

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. I low many owned neither IBM nor GE?

e. I low many owned either IBM or AT&T but no GE?

To determine

To find:

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?

Answer to Problem 29AYU

a. 15.

b. 15.

c. 15.

d. 25.

e. 40.

Explanation of Solution

Given:

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.

Formula used:

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

$n(\cup )=n\left(A\cup B\right)+n\left(\overline{A\cup B}\right)$

Calculation:

From the given data, we draw the venn diagram as shown below.

$n\left(\text{IBM}\right)=50$

$n\left(\text{AT }\&\text{ T}\right)=40$

$n\left(\text{GE}\right)=45$

$n\left(\text{IBM }\cap \text{ GE}\right)=20$

$n\left(\text{AT }\&\text{ T }\cap \text{ GE}\right)=15$

$n\left(\text{IBM }\cap \text{ AT }\&\text{ T}\right)=20$

$n\left(\text{IBM }\cap \text{ AT }\&\text{ T }\cap \text{ GE}\right)=5$

$n\left(\left(\text{IBM }\cap \text{ GE}\right)\text{ only}\right)=20-5=15$

$n\left(\left(\text{IBM }\cap \text{ AT }\&\text{ T}\right)\text{ only}\right)=20-5=15$

$n\left(\left(\text{AT }\&\text{ T }\cap \text{ GE}\right)\text{ only}\right)=15-5=10$

$n\left(\text{IBM only}\right)=50-15-15-5=15$

$n\left(\text{AT }\&\text{ T only}\right)=40-15-10-5=10$

$n\left(\text{GE only}\right)=45-10-15-5=15$

From the venn diagram,

a. Number of the investors surveyed did not have shares in any of the three companies is 15.

b. Number of investors who owned just IBM shares are 15.

c. Number of investors who owned just GE shares are 15.

d. Number of investors who owned neither IBM nor GE is $n\left(U\right)+n\left(\text{AT }\&\text{ T only}\right)=15+10=\text{ 2}5$.

e. Number of investors who owned either IBM or AT&T but no GE is $n\left(\text{IBM only}\right)+n\left(\text{AT }\&\text{ T only}\right)+n\left(\text{IBM and AT }\&\text{ T}\right)=15+15+10=40$.