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 28% of ice cream sales are vanilla, Chocolate accounts for only 11% of ice cream sales. Suppose that 183 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 ill 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? Hint: Chocolate flavor and vanilla flavor are not mutually exclusive events. Assume that the choice to buy one flavor is independent of the choice to buy another flavor. Then use the multiplication rule for independent events, together with the addition rule for events that are not mutually exclusive, to compute the requested probability. (d) What is the probability that between 50 and 60 customers will buy chocolate or vanilla ice cream? Hint: Use the probability of success computed in part (c).

Answer 1

a) X ~ Binornial(n = 183, p = 0.28) Binomial can be approximated to normal with: μ-np-183 * 0.28-51.24 σ-v/rp(1 _ p)-V/183 * (0.28) (1-0.28) 6.073944 6.074 P(X 2 50) Since we are approximating a discrete distribution by continuous normal distri bution, values between 49.5 and 50.5 both approrimate to 50. Thus, "greater than or equal to 50 corresponds to continuous normal distribution with X > 49.5 after continuity coTrection. Normal Distribution, x-49.5, μ-51.24, σ-6.074 We convert this to standard normal using z- 49.5 51.24 6.074 ะ-0.28647 ะ-0.29 P(X > 49.5) - Area to the right of 49.5 μ = 51.24 P(X >49.5 - 6.073944 49.5 51.24 P(X > 49.5)-P(Z >-0.29) = 1-P(Z <-0.29) = 1-0.3859 (from z-table) 0.6141 = P(X > 50) = 0.6141 Using excel instead of z-table gives: 0.6127407436 Note that if this question is solved without continuity correction, the answer would have been P(X > 50)- P(Z > -0.20)0.5793

Without integer-continuity correction:P(X50)-0.5793 Using excel instead of z-table gives: 0.5808821374 b) X ~ Binornial(n-183, p-0.11) Binomial can be approximated to normal with: μ = np = 183 * 0.11 = 20.13 ơ-v/rp(1 _ p)-V183 * (0.11) (1-0.11) 4.232694 ~ 4.233 P(X 2 12) Since we are approximating a discrete distribution by continuous normal distri bution, values between 11.5 and 12.5 both approrimate to 12. Thus, "greater than or equal to 12 " сотт℃sponds to continuous normal distribution with X > 11.5 after continuity coTrection. Normal Distribution, z = 11.5, μ = 20.13, σ = 4.233 We convert this to standard normal using z- 115-20.13 4.233 ~一2.038891 ~-2.04 P(X > 11.5) = Area to the right of 11.5 μ-2013 σ 4.232694 X 11.5 11.5 20.13 P(X > 11.5)-P(Z >-2.04) = 1-P(Z <-2.04) = 1-0.0207 (from z-table) 0.9793 = P(X > 12)-0.9793 Using excel instead of z-table gives: 0.9792695263

Note that if this question is solved without continuity correction, the answer would have been P(X > 12)- P(Z>-1.92) 0.9726 Without integer-continuity correction:P(X > 12)-0.9726 Using excel instead of z-table gives: 0.9726191733 c) P(chocolate or vanila) 0.280.11 0.28*0.110.3592 d) X ~ Binomial (n-183, p-0.3592) Binomial can be approximated to normal with: μ-np-183 * 0.3592-65.733600 65.734 σ v/rp(1 _ p)-V 183 * (0.3592) (1-0.3592) 6.4901 53 6.490 P(50 X < 60) Since we are approrimating a discrete binomial distribution by continuous nor- mal distribution, values between 49.5 and 50.5 both approzimate to 50. Thus "between 50 and 60" corresponds to continuous normal distribution with P (49.5< X<60.5) after continuity correction Normal Distribution, xi-49.5,#2-60.5, μ-65.734, σ= 6.490 We convert this to standard normal using z- 49.5 65.734 6.490 60.5-65.734 6.490 ~-2.50 1266 ~-2.50 -0.806391-0.81 P(49.5< X < 60.5) P-2.50< Z< -0.81) = P(Z <-0.81)-P(Z <-2.50) 0.2090 0.0062 (from z-table) -0.2028 P(50 < X < 60) 0.2028 Using excel instead of z-table gives: 0.2038212533 Without continuity correction, answer is: 0.1808313020