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.)
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.
To solve this problem, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k customers who report favorably
- n is the number of customers surveyed
- p is the probability of a customer reporting favorably
- (n C k) is the combination formula for choosing k items from n items
Given that n = 50 and p = 0.78, let's solve the problems.
a. To find the probability that the dealership will be fined, we need to find the probability that the number of customers who report favorably is between 32 and 34. In other words, we need to find P(32 <= X <= 34).
Using the binomial probability formula, we can calculate:
P(32 <= X <= 34) = P(X = 32) + P(X = 33) + P(X = 34)
P(X = k) = (50 C k) * 0.78^k * (1-0.78)^(50-k)
P(32 <= X <= 34) = (50 C 32) * 0.78^32 * (1-0.78)^(50-32) + (50 C 33) * 0.78^33 * (1-0.78)^(50-33) + (50 C 34) * 0.78^34 * (1-0.78)^(50-34)
Using a calculator or a spreadsheet, you can calculate this probability. The result should be rounded to 4 decimal places.
b. To find the probability that the dealership will be dissolved, we need to find the probability that fewer than 32 customers report favorably. In other words, we need to find P(X < 32).
Using the binomial probability formula, we can calculate:
P(X < 32) = P(X = 0) + P(X = 1) + ... + P(X = 31)
P(X = k) = (50 C k) * 0.78^k * (1-0.78)^(50-k)
P(X < 32) = P(X = 0) + P(X = 1) + ... + P(X = 31)
Using a calculator or a spreadsheet, you can calculate this probability. The result should be rounded to 4 decimal places.
To calculate the probabilities, we need to use the binomial distribution formula, which is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the number of combinations of n items taken k at a time,
- p is the probability of a single success, and
- n is the number of trials.
a. To find the probability that the dealership will be fined (between 32 and 34 customers reporting favorably), we need to find the cumulative probability of 32, 33, and 34 customers reporting favorably and subtract it from 1.
Step 1: Calculate the cumulative probability of fewer than 32 customers reporting favorably.
P(X < 32) = 1 - P(X >= 32)
P(X < 32) = 1 - (P(X = 32) + P(X = 33) + P(X = 34))
Step 2: Calculate the probabilities of 32, 33, and 34 customers reporting favorably.
P(X = k) = C(50, k) * (0.78^k) * (1-0.78)^(50-k)
P(X = 32) = C(50, 32) * (0.78^32) * (1-0.78)^(50-32)
P(X = 33) = C(50, 33) * (0.78^33) * (1-0.78)^(50-33)
P(X = 34) = C(50, 34) * (0.78^34) * (1-0.78)^(50-34)
Step 3: Calculate the cumulative probability of 32, 33, and 34 customers reporting favorably.
P(X < 32) = 1 - (P(X = 32) + P(X = 33) + P(X = 34))
b. To find the probability that the dealership will be dissolved (fewer than 32 customers reporting favorably), we need to calculate the cumulative probability of fewer than 32 customers reporting favorably.
P(X < 32) = 1 - P(X >= 32)
Now, you can substitute the values into the formulas and use the z-table to find the values of C(n, k) for each case.