In the following problem, check that it is appropriate to use the normal approxi

Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. What's your favorite ice cream flavor? For people who buy ice cream, the all- time favorite is still vanilla. About 29% of ice cream sales are vanilla. Chocolate accounts for only 8% of ice cream sales. Suppose that 174 customers go to a grocery store in Cheyenne, Wyoming, today to buy ice cream. (Round your answers to four decimal places.) (a) What is the probability that 50 or more will buy vanilla? (b) What is the probability that 12 or more will buy chocolate? (c) 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 vanilla? (d) What is the probability that between 50 and 60 customers will buy chocolate or vanilla ice cream?

Explanation / Answer

a ) P (vanilla) = 0.29 , n = 174

mean = n p = 174 * 0.29 = 50.46

std.deviation = sqrt ( npq) = sqrt ( 50.46 * 0.71) =5.98

P (x > 50)

z = ( x - mean) / s

= ( 50 - 50.46) / 5.98

= -0.077

Now, we need to find p(z >-0.077)

P(X >50 ) = p(z >-0.077)= 0.5307

b)

P (vanilla) = 0.08 , n = 174

mean = n p = 174 * 0.08 = 13.92

std.deviation = sqrt ( npq) = sqrt ( 13.92 * 0.92) = 3.578

P (x > 12)

z = ( x - mean) / s

= ( 12 - 13.92) / 3.578

= -0.536

Now, we need to find p(z >-0.536)

P(X >12 ) = p(z >-0.536)= 0.7042

c)

Probability that customer buys vanilla, P(V) = 0.29

Probability that customer buys Chocolate, P(C) = 0.08

P(V and C) = P(V) * P(C) = 0.29*0.08 = 0.0232

Required probability P(V or C) = P(V) + P(C) - P(V and C) = 0.29 + 0.08 - 0.0232 = 0.3468

(D)

p = 0.3468, n = 174

mean = n p = 174 * 0.3468 = 60.3432

std.deviation = sqrt ( npq) = sqrt ( 174 * 0.3468 * 0.92) = 7.4509

P (50 > x > 60) = P((50-60.3432)/7.4509 < z < (60-60.3432)/7.4509) = P( -1.3881 < z < -0.0461) = 0.3991

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.