Answer to Question #299637 in Statistics and Probability for Nani

Question

A company study of product preferences of 10,000 consumers is as follows: products A. B and C were liked by 5015, 3464 and 4827 consumers, respectively. All the three products were liked by 500 consumers. Products A and B were liked by 1000 consumers. Products A and C were liked by 850 consumers. Products B and C were liked by 1420 consumers.

1) Prove that the study results are incorrect.

2)It was found that an error occurred in recording the number of consumers liking the products A and C. What is the value of this number?

Answer 1

Total number of consumers is 10000

Product A liked by 5015 consumers

Product B liked by 3464 consumers

Product C liked by 4827 consumers

All three products liked by 500 consumers.

Products A and B were liked by 1000 consumers

Products B and C were liked by 1420 consumers

We can represent this information in the venn diagram below.

Number of consumers who liked product A only is 5015-500-500-350=3665

Number of consumers who liked product B only is 3464-500-500-920=1544

Number of consumers who liked product C only is 4827-500-350-920=3057

Number of consumers who liked exactly 1 product is 3665+1544+3057=8266....(1)

Number of consumers who liked exactly 2 products is 350+500+920=1770.........(2)

Number of consumers who liked all the three products is 500........(3)

If we take the sum of equations (1),(2) and (3) above, we should get the total number of consumers

Summing gives,

$8266+1770+500=10536\not=10000$

Therefore, the study results are incorrect.

2)

The value of the error is, 10536-10000=536

The number of consumers who liked products A and C is 850. So, we need to subtract 536 from 850 to get the correct number of consumers who liked products A and C.

Now, 850-536=314

Therefore, the correct number of consumers who liked products A and C is 314