Question

A department store surveyed 428 shoppers, and the following information was obtained: 216 shoppers made a purchase, and 294 were satisfied with the service they received. If 47 of those who made a purchase were not satisfied with the service, how many shoppers did the following?a. made a purchase and were satisfied with the service
b. made a purchase or were satisfied with the serice
c. were satisfied with the service but did not mak a purchase
d. were not satisfied and did not make a purchase

Answer 1

Answer:

(a) 169

(b) 341

(c) 125

(d) 87

Step-by-step explanation:

Consider the Venn diagram below.

The total number of shoppers surveyed is, N = 428.

Number of shoppers who made a purchase, n (P) = 216

Number of shoppers who were satisfied with the service they received,

n (S) = 294

Number of shoppers who made a purchase but were not satisfied with the service, $n(S^{c}\cap P)$ = 47

(a)

The number of shoppers who made a purchase and were satisfied with the service = n (S ∩ P)

             $n(S\cap P)=n(P)-n(S^{c}\cap P)\\=216-47\\=169$

(b)

The numbers of shoppers who made a purchase or were satisfied with the service = n (P ∪ S)

               $n(P\cup S)=n(P)+n(S)-n(P\cap S)\\=n(P)+n(S)-n(S\cap P)\\=216+294-169\\=341$

(c)

The numbers of shoppers who were satisfied with the service but did not make a purchase = $n(S\cap P^{c})$

$n (S\cap P^{c} ) = n (S) - n (S \cap P)\\=294-169\\=125$

(d)

The number of shoppers who were not satisfied and did not make a purchase  = $n(S^{c}\cap P^{c})$

$n(S^{c}\cap P^{c})=N-n(S\cup P)\\=N-n(P\cup S)\\=428-341\\=87$