the r cream sales. Suppose that 167 customers go to a grocery store in Cheyenne,

Question

the r cream sales. Suppose that 167 customers go to a grocery store in Cheyenne, Wyoming, today to buy ice cream. (Round your answers to four decimal places.) What is the probability that 12 or more will buy chocolate? (e) A customer who buys ice cream is not limited to one container or one flavor. What is the probability that someone who is buying ice cream will buy chocolate or vanila? Hint: events. Assume that the choice to buy one flavor is independent of the choice to buy another flavor. Then use the rule for independent events, together with the addition rule for events that are not mutually exclusive, to compute the requested probability

Explanation / Answer

Here n = 167

p (vanilla) = pv = 0.20

p(chocolate) = pc = 0.12

(a) Here as sample size is quite large we can use normal appoximation to binomial here.

Here expected number of vanilla users = 0.20 * 167 = 33.4

standard deviation of number of vanilla users = sqrt[0.20 * 0.80 * 167 ] = 5.17

Pr(X >= 50; 33.4 ; 5.17) = 1 - Pr(X< 50 ; 33.4 ; 5.17)

Z = (50 - 33.4)/ 5.17 = 3.21

Pr(X >= 50; 33.4 ; 5.17) = 1 - Pr(X< 50 ; 33.4 ; 5.17) = 1- Pr (Z < 3.21)

= 1 - 0.9993 = 0.0007

(b) Expected number of chocolate users = 167 * 0.12 = 20.04

standard deviation of number of chocolate users = sqrt[0.12* 0.88 * 167 ] = 4.20

Pr(X >= 12; 20.04 , 4.20) = 1 - Pr(X< 12; 20.04 ; 4.20)

Z = (12 - 20.04)/4.20 = -1.91

Pr(X >= 12; 20.04 ; 4.20) = 1 - Pr(X < 12; 20.04 ; 4.20)   = 1- Pr (Z < -1.91)

= 1 - 0.0281 = 0.9719

(c) Pr (someone who is buying icecream can buy vanilla or chocolate) = ?

as both events are not mutually exclusive.

Pr (someone who is buying icecream can buy vanilla or chocolate) = Pr(vanilla) + Pr(Chocolate) - Pr(vanilla or chocolate) = 0.2 + 0.12 - 0.2 * 0.12 = 0.296

(d) Pr(50 <= X <= 60) = Pr(X <= 60) - Pr(X <= 50)

Here expected number of customers who buy chocolate or vanilla = 167 * 0.296 = 49.43

Standard deviation of number of customers who can buy chocolate or vanilla = sqrt [0.296 * 0.704 * 167] = 5.9

so Pr(50 <= X <= 60) = Pr(X <= 60 ; 49.43; 5.9) - Pr(X <= 50; 49.43; 5.9) = (Z2 ) - (Z1)

Z2 = (60 - 49.43)/ 5.9 = 1.79 ; Z1 = ( 50 - 49.43)/ 5.9 = 0.10

Pr(50 <= X <= 60) = Pr(X <= 60 ; 49.43; 5.9) - Pr(X <= 50; 49.43; 5.9) = (Z2 ) - (Z1) = (1.79) - (0.10)

= 0.9633 - 0.5398 = 0.4235

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