To solve this problem, we can use the binomial distribution formula.
a. To find the probability that the dealership will be fined, we need to find the probability of having between 32 and 34 customers who report favorably.
P(32 ≤ X ≤ 34) = P(X = 32) + P(X = 33) + P(X = 34)
We can use the binomial distribution formula to calculate each probability. The formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where n is the number of trials, k is the number of successful outcomes, p is the probability of success, and C(n, k) is the number of combinations.
Using the given information, n = 50, k = 32, and p = 0.78.
P(X = 32) = C(50, 32) * 0.78^32 * (1-0.78)^(50-32) = 0.0573
P(X = 33) = C(50, 33) * 0.78^33 * (1-0.78)^(50-33) = 0.1104
P(X = 34) = C(50, 34) * 0.78^34 * (1-0.78)^(50-34) = 0.1539
Therefore, P(32 ≤ X ≤ 34) = 0.0573 + 0.1104 + 0.1539 = 0.3216
The probability that the dealership will be fined is 0.3216.
b. To find the probability that the dealership will be dissolved, we need to find the probability of having fewer than 32 customers who report favorably.
P(X < 32) = P(X = 0) + P(X = 1) + ... + P(X = 31)
We can use the binomial distribution formula to calculate each probability.
P(X < 32) = P(X = 0) + P(X = 1) + ... + P(X = 31) = 1 - P(X = 32) - P(X = 33) - P(X = 34)
P(X < 32) = 1 - 0.0573 - 0.1104 - 0.1539 = 0.6784
The probability that the dealership will be dissolved is 0.6784.
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