To solve this problem, we can use the binomial distribution formula. Let's define the following variables:
n = number of trials (number of customers surveyed) = 50
p = probability of success (customers reporting favorably) = 0.78
a. To find the probability that the dealership will be fined (number of customers reporting favorably between 32 and 34), we need to calculate the cumulative probability of the number of successes falling within this range.
P(32 ≤ x ≤ 34) = P(x = 32) + P(x = 33) + P(x = 34)
Using the binomial distribution formula, we can calculate the probability of each specific outcome:
P(x = 32) = (50 C 32) * (0.78)^32 * (1-0.78)^(50-32)
P(x = 33) = (50 C 33) * (0.78)^33 * (1-0.78)^(50-33)
P(x = 34) = (50 C 34) * (0.78)^34 * (1-0.78)^(50-34)
Next, we can calculate these probabilities and sum them up to find the probability of the dealership being fined:
P(dealership fined) = P(32 ≤ x ≤ 34) = P(x = 32) + P(x = 33) + P(x = 34)
Now, let's calculate the probabilities:
P(x = 32) = (50 C 32) * (0.78)^32 * (1-0.78)^(50-32) = 0.1257 (rounding to 4 decimal places)
P(x = 33) = (50 C 33) * (0.78)^33 * (1-0.78)^(50-33) = 0.1725 (rounding to 4 decimal places)
P(x = 34) = (50 C 34) * (0.78)^34 * (1-0.78)^(50-34) = 0.1697 (rounding to 4 decimal places)
P(dealership fined) = P(32 ≤ x ≤ 34) = P(x = 32) + P(x = 33) + P(x = 34) = 0.1257 + 0.1725 + 0.1697 ≈ 0.4679 (rounding to 4 decimal places)
Therefore, the probability that the dealership will be fined is approximately 0.4679 or 46.79%.
b. To find the probability that the dealership will be dissolved (fewer than 32 customers report favorably), we can calculate the cumulative probability of the number of successes being less than 32.
P(x < 32) = P(x = 0) + P(x = 1) + ... + P(x = 31)
Using the binomial distribution formula, we can calculate the probability of each specific outcome and sum them up to find the probability of the dealership being dissolved:
P(dealership dissolved) = P(x < 32) = P(x = 0) + P(x = 1) + ... + P(x = 31)
Now, let's calculate the probabilities:
P(x = 0) = (50 C 0) * (0.78)^0 * (1-0.78)^(50-0) = 0.0000 (rounding to 4 decimal places)
P(x = 1) = (50 C 1) * (0.78)^1 * (1-0.78)^(50-1) = 0.0001 (rounding to 4 decimal places)
... (calculating probabilities for x = 2 to x = 31) ...
P(x = 31) = (50 C 31) * (0.78)^31 * (1-0.78)^(50-31) = 0.0882 (rounding to 4 decimal places)
P(dealership dissolved) = P(x < 32) = P(x = 0) + P(x = 1) + ... + P(x = 31)
Calculating the sum of the probabilities:
P(dealership dissolved) = P(x < 32) = 0.0000 + 0.0001 + ... + 0.0882 ≈ 0.0781 (rounding to 4 decimal places)
Therefore, the probability that the dealership will be dissolved is approximately 0.0781 or 7.81%.
A car manufacturer is concerned about poor customer satisfaction at one of its dealerships. The management decides to evaluate the satisfaction surveys of its next 50 customers. The dealership will be fined if the number of customers who report favorably is between 32 and 34. The dealership will be dissolved if fewer than 32 customers report favorably. It is known that 78% of the dealership’s customers report favorably on satisfaction surveys. [You may find it useful to reference the z table.]
a. What is the probability that the dealership will be fined? (Round final answer to 4 decimal places.)
b. What is the probability that the dealership will be dissolved? (Round final answer to 4 decimal places.)
1 answer