Question

A large grocery store chain believes that 78​% of its customers use the​ store's "Savings​ Card" when they make purchases. The manager in charge of the​ "Savings Card" program has randomly sampled 23 customers and observed whether or not they used the card. Complete parts a through c. a. Assuming that the 78​% use rate is​ correct, what is the probability that 18 or more people in the sample used the​ card?

Answer 1

The probability that 18 or more people in the sample used the card is P=0.6049.

Explanation:

This question can be solved using the binomial distribution, with size n=23 and p=0.78.

The probability that 18 out of 23 people uses the card is:

`P(X\geq18)=\sum_(18)^(23)P(i)\\\\\\P(18)=\binom{23}{18}p^18(1-p)^5=33649\cdot0.0114\cdot0.0005=0.1981\\\\P(19)=\binom{23}{19}p^19(1-p)^4=8855\cdot0.0089\cdot0.0023=0.1848\\\\P(20)=\binom{23}{20}p^20(1-p)^3=1771\cdot0.0069\cdot0.0106=0.131\\\\P(21)=\binom{23}{21}p^21(1-p)^2=253\cdot0.0054\cdot0.0484=0.0664\\\\P(22)=\binom{23}{22}p^22(1-p)^1=23\cdot0.0042\cdot0.22=0.0214\\\\P(23)=\binom{23}{23}p^23(1-p)^0=1\cdot0.0033\cdot1=0.0033\\\\\\P(X\geq18)=0.1981+0.1848+0.131+0.0664+0.0214+0.0033=0.6049`

The probability that 18 or more people in the sample used the card is P=0.6049.